The ledger

What the figures prove — page 4

Page 4 of 6, continuing through the fields in the same order. Every claim here was checked as the drawing was made rather than typed beside it.

• |qα − p| comes out below 1/N
• and every convergent from the first onward breaks a record
• every record-breaking fraction is a convergent
• neighbouring convergents differ by a determinant of one
• so p/q is within one over qN of the number itself
• the constant is one this figure knows
• the largest denominator is a whole number between 8 and 200
• the number of boxes is a whole number between 3 and 14
• there are record-breaking denominators to mark
• two of the N + 1 points share a box, as they must

• the period of √5 is odd, so x² − 5y² = −1 has a solution ×211
• the classes of x² − 2y² = 1 counted in the box agree with the count from 1's factorisation ×200
• term 0 of e follows Euler's pattern ×40
• term 0 of the continued fraction of ∛2 is 1 ×29
• term 0 agrees with the decimal value ×12
• term 0 of π is 3 ×11
• q² times the error of convergent 1 ×8
• the class number of Q(√5) is 1 ×7
• the classes walked out with the unit find every solution of x² − 2y² = 1 in range ×6
• √34's cycle is as long as its period ×5
• and its last term is twice the first term 5 ×5
• the cycles of 34 use every reduced state once ×5
• the period of √29 before its last term reads the same backwards ×5
• the quotients rebuild 34/13 ×3
• D is a whole number from 2 to 1000 and not a square ×2
• D runs up to between 20 and 300 ×2
• the number of terms is a whole number between 2 and 12 ×2
• x² − 2y² = 7 has a solution to draw ×2
• √2 has a convergent above √5, as Hurwitz says every irrational must
• √D's own states are exactly one of the cycles, as long as its period
• 4657 first divides y at the 2329th power, a divisor of 4657 + 1
• 4729494 is not a square modulo 4657, which is why the order divides 4657 + 1
• a product of solutions for N₁ and N₂ solves N₁N₂
• a row of −1 composed with itself gives a row of 1
• a step from a reduced state lands on a reduced state
• after the first step |k| is below √D
• after the first step |k| stays below √D
• an odd period never occurs when a prime 4k + 3 or 4 divides D
• and for primes 4k + 3 it is even
• and needs fewer steps than the convergents do
• and that row is the method's own answer
• and the first that does is the fundamental solution found by searching
• and the last of them has closed on √5
• between four and twelve terms are expanded
• between two and five constants
• both right-hand sides have solutions
• cols is read only by the negpell view
• count is read only by the cfcubic and gausskuzmin and khinchin and cfapprox view
• D is a whole number between 2 and 1000 and not a perfect square
• D is between 2 and 40
• D is not a perfect square
• D is not a perfect square, or the equation has only the trivial solution
• D names the Pell equation and is read only by the pell, chakravala, compose, states, reduced, nclasses, nagell, nproduct and quotients views; the other views take `of`
• dilations is between 2 and 6 and is read only by the dilate and ehrhart views
• dividing by k gives the method's next row
• dividing out k leaves whole numbers
• Ds is a short list of equations and is read only by the kwalk, palindrome and cycles views
• e has a convergent above √5, as Hurwitz says every irrational must
• each convergent is closer than the one before
• enough digits to fix forty terms of π and of e exactly
• every class number here is odd, as genus theory requires
• every constant expanded is one this family knows
• every convergent stays close to the hyperbola
• every period up to 1,000 is a palindrome ending in twice the first term
• every point walked out is a solution
• every prime of the form 4k + 1, and 2, gives a solvable equation
• every product lands in one of the classes the box finds
• every reduced state is above 1 with its conjugate between −1 and 0
• every reduced state lies on a cycle
• every row is a near miss a² − Db² = k
• from and to bound a run of nine to twenty-five consecutive D
• from is read only by the neighbours view
• N is a whole number from 1 to 400
• N is read only by the nagell view
• neighbouring convergents differ by a determinant of one
• Ns is a list of two to seven right-hand sides
• Ns is read only by the nclasses view
• pair is read only by the nproduct view
• pair is two coprime right-hand sides from 2 to 40
• reduction carries reduced forms to reduced forms
• rs is a short list of whole heights and is read only by the reeve and ehrhart views
• scale is digits or root
• scale is read only by the sizes view
• some convergent solves the equation exactly
• span is read only by the nclasses view
• the 2 × 2 products land in 4 different classes, every one
• the class number formula h·R = √p·L(1, χ)/2 holds with the computed regulator
• the composition multiplies the two misses
• the constant is one this figure knows
• the convergents alternate above and below the value
• the cyclic method finishes
• the cyclic method needs fewer steps than the convergents
• the cyclic method reaches the continued fraction's fundamental solution
• the denominator is a whole number between 1 and 10000000
• the denominators grow at close to Lévy's rate
• the error view compares a list of constants and every other view expands one
• the expansion closes a period and finds the solution
• the expansion repeats
• the first prime p ≡ 1 mod 4 with class number above one is 229
• the fundamental solution of x² − 4729494y² = 1 solves it
• the fundamental solution solves the equation
• the golden ratio's convergents never leave the neighbourhood of √5
• the law's shares are computed
• the method takes at least one step
• the numerator is a whole number between 1 and 10000000
• the period is never more than the number of reduced states
• the polynomial changes sign between the integer part and one more
• the product has norm +1 when the period is even
• the product has norm −1 when the period is odd
• the product is the fundamental solution of x² − 94y² = 1
• the product of one period's complete quotients has whole-number coordinates
• the product, squared, is the fundamental solution of x² − 61y² = 1
• the recurrence stays in whole numbers
• the running geometric mean is near Khinchin's constant
• the share of 1s is close to the law's 41.5%
• the share so far lies above Stevenhagen's limit and has fallen since the start
• the share with class number one sits above Cohen and Lenstra's prediction and drifts towards it
• the solution found satisfies the equation
• the stacked logarithms add up to ln of the product
• the strip holds between two and eight steps of the unit
• the tower's last convergent is a number
• the unit sends a solution to a solution
• the view is one the family draws
• there are enough convergents to draw
• to is read only by the neighbours view
• upTo is a multiple of ten from 30 to 120
• upTo is read only by the chakcount, periods, ncount, sizes and five negative-Pell views
• when both are +1, some are and some are not
• when both fourth-power symbols are −1, every one is (Scholz)
• when p is not a square modulo q, every product is solvable
• when the fourth-power symbols differ, none is (Scholz)
• y₁ is already even, so 9314 = 2 · 4657 divides y exactly when 4657 does
• π has a convergent above √5, as Hurwitz says every irrational must
• π has a convergent that beats √5 by two orders of magnitude
• φ has a convergent above √5, as Hurwitz says every irrational must

• the side of the big square is a whole number between 2 and 200 ×2
• the side of the small squares is a whole number between 1 and 200 ×2
• and the discrepancy never changes size
• and the new pair is strictly smaller
• every step is strictly smaller than the one before
• the descent carries the discrepancy to minus itself
• the descent runs out of room before the discrepancy runs out
• the descent takes at least three steps
• the pair is one the descent can start from
• the two small squares overlap and still reach the corners
• the two small squares, less their overlap and plus the corners, are the big one

• every factorisation of 5 has the same number of factors ×4999
• the rule and the search agree at 1 ×3000
• the divisor sum of 12 from its factorisation matches the list ×1635
• n² + n + 2 is prime for every n up to 0 exactly when 7 is on the list ×78
• 1 has no friend below the bound ×51
• n² + n + 41 is prime at n = 0 ×40
• 9 is irreducible because it is two primes of the form 4k + 3 ×32
• 5 fails Euclid's lemma exactly when it is not an ordinary prime ×28
• 1/24 is a sum of distinct divisors over 24 ×23
• 6 has abundancy 2/1 ×23
• the unit fractions for 1/24 add back to it ×23
• 6 is called what it is ×10
• the bar for 6 is its divisors laid end to end ×10
• weird number 1 ×10
• 70's excess cannot be made from its small divisors ×9
• class number one at −3 ×9
• the discs cover the plane exactly for the five Euclidean fields (1) ×9
• the lattice for −1 leaves points at most 0.707 from it ×9
• 2 to the 0 divides every class number with 1 primes in the discriminant ×6
• every known 2-perfect number has enough primes ×5
• the count for 2 is found ×5
• the lattice draws every divisor of 30 exactly once ×5
• σ(n) is exactly 2 times n ×5
• ω(12) ×5
• Ω(12) ×5
• the range is a whole number between 50 and 20000 ×4
• 9 cannot be split into two numbers of the form 4k + 1 ×3
• 9 is a Hilbert number ×3
• the factors multiply to 3 ×3
• the sieve agrees with the reduced forms at −23 ×3
• 3 leaves remainder 3 on division by 4 ×2
• 496 is perfect ×2
• the bound is a whole number between 100 and 3000 ×2
• the number is a whole number between 4 and 100000 ×2
• the σ of the pieces multiply to 3 times n ×2
• (a, −b, c) is the inverse of (a, b, c)
• 1,000 to 200,000
• 10 is the first number of unknown status
• 12 is practical by both tests
• 18,518,809 is 197² + 1848·100²
• 24 is practical
• 2x² + 2xy + 3y² takes exactly the primes 3 or 7 mod 20
• 3, 5 and 7 divide the class number more often than one time in p
• 441 has exactly two factorisations into Hilbert irreducibles
• 441 is 3 × 3 × 7 × 7
• 5/4 is ruled out: its denominator forces n even
• 6 and 220 are classified as what they are
• 6 and 28 sit on the line
• 60, 90, 87,360 or Wall's number
• 70 is abundant: its proper divisors add to 74
• a 3-part of order nine is cyclic or the square of the group of order three
• a held term leads to a term at least as large that is again held by the perfect number
• a negative discriminant, 0 or 1 mod 4
• a number from 6 to 1,000
• a prime q divides p − 1 for a share 1/(q − 1) of primes
• a prime this form takes, it takes once
• a proper multiple of a perfect or abundant number is abundant
• a random permutation of 100 has no cycle longer than half with chance near 1 − log 2
• a square root of −1 modulo the prime
• a table of 12 to 40 rows
• a table of 300, 1,000 or 2,000 rows
• a term held by the perfect driver never shrinks
• a visible share of the range sits just above one
• an even exponent gives an odd divisor sum
• an idoneal class number is a power of two
• an odd exponent gives 2 mod 4 exactly when p and e are both 1 mod 4
• an odd perfect number needs at least three distinct primes
• and associative
• and at n = 40 it is 41²
• and so do the norms of two and three
• and the next term is again the perfect number times an odd number
• and the two forms take the same values
• and they are the same numbers
• at 10⁷ the counted law is within seven points of Dickman's at every level, and closer than at 10⁴
• at ten million the spread is far narrower than the limiting bell
• at this range more fields have no class of order 3 than the limit predicts
• below 1000, the sequences that pass 10^22 are the twelve undecided ones and 840
• below a million only 10 itself has abundancy 9/5
• below ten million the unitary perfect numbers are 6, 60, 90 and 87,360
• between one and eight whole numbers above one
• between one and five starting values from 2 to ten million
• both first splits divide the number
• both trees end in the same multiset of primes
• but 22021 = 19²·61
• composition is commutative
• Descartes' number is 198,585,576,189
• discriminant −20 has two classes
• each average closes in on log log plus its own constant
• each step keeps the discriminant
• every cycle of a permutation of n has length at most n
• every divisor including the number itself adds to twice the number
• every even number up to 4000 is a sum of two practical numbers
• every one of these monoids has an element with two factorisations
• every product is one of the reduced forms
• every reduced form's root lies in the region
• every run share lies on the drawn axis
• every sequence is decided or passes the bound within the step budget
• every total from 1 to 28 is a sum of distinct divisors of 12
• every weird number in range is only slightly abundant: its excess is under a tenth of it
• for 4k + 1 it is 441
• for every odd n, 2 to the power of its number of primes divides σ*(n)
• in fact at most half the divisor's norm, because the square lattice is covered by discs of radius √2/2
• log log N reaches 3 at about 10⁸·⁷
• longer runs of smooth neighbours fall further below the independent rate
• n + 1 has the larger largest prime about half the time
• neighbours are both smooth less often than independent numbers
• no divisor is drawn twice
• no element of the ring has norm 2 or 3
• no element of this ring has norm two or three, so neither factor can split further
• no idoneal number between 1,849 and the bound
• no odd number below the bound is perfect
• no other fundamental discriminant in range has class number one
• no prime is taken by both forms
• no set of divisors adds to 70, because none adds to the excess 4
• numbers in the table have more prime factors on average than numbers below n² in general
• odd abundant numbers in range that are no sum of their divisors
• odd weird numbers in range
• one primitive representation and nothing else: prime, as Euler's rule says
• q divides n² + 1 for a share ρ(q)/q of n
• rank one is common and rank two is rare but present
• remainders decide exactly when n is idoneal
• rounding the exact quotient leaves a remainder of smaller norm
• the average is log log N plus Mertens's constant, to within a twentieth
• the average of |Cl[3]| rises towards 2 from below
• the average of Ω − ω is the sum over primes of 1/(p(p − 1))
• the cells add up to the divisor sum
• the class number is odd exactly when the discriminant has one prime factor
• the class number is on average a fixed fraction of √|D|
• the class-number-one discriminants are the nine Heegner found
• the classes of order dividing 3 form a group of size a power of 3
• the composed form has whole coefficients
• the corner where both neighbours are nearly prime is empty, though independence would fill it
• the corrected share stays between fixed bounds over the range
• the correlation is negative at distance one and positive at distances divisible by six
• the count of primes n² + 1 matches the prediction within two per cent
• the counts stay above the limit
• the divisor count is the product of one more than each exponent
• the divisor sum factorises as the two row totals multiplied
• the driver actually holds somewhere on the drawn sequence
• the driver is one of the first four perfect numbers
• the entries are counted once each
• the exponent is a whole number between 2 and 7
• the first discriminant with two independent classes of order 3 is −3299, class number 27
• the first odd abundant number is 945
• the first step shown is a whole number between 0 and 200
• the first tree's leaves multiply back to the number
• the Hardy–Littlewood constant for n² + 1 is 1.3728
• the idoneal numbers up to the bound are exactly Euler's sixty-five
• the largest denominator is a whole number between 4 and 16
• the largest number is a whole number between 101 and 600
• the largest pairs for 3, 5 and 7 are 8 and 9, 80 and 81, 4374 and 4375
• the last step shown is a whole number between 10 and 37
• the lattice half-width is a whole number between 2 and 5
• the mean falls towards the Golomb–Dickman constant
• the mean is still above its limit at 10⁷
• the Mersenne number for this exponent is prime
• the model gives a real chance to counts no number this small can have
• the model is more than twice as spread out as the truth
• the model's average matches the true average
• the model's chances add to one
• the nearest are at distance √(5/4), so every remainder has norm 4 × 5/4 = 5, more than N(2) = 4
• the number has a factor tree to draw
• the number has at least three prime factors, so its trees can differ
• the number has at most three distinct primes
• the number is a product of exactly two prime powers
• the number is inside the range the primality test decides
• the numbers with no repeated prime are 6/π² of all
• the only odd m with an odd, coprime completing factor is Descartes'
• the option is one this view reads
• the perfect numbers below a million form one club of four
• the powers of two below 2^k add up to the Mersenne number
• the practical count at the top sits within a hundredth of 1.336 x / log x
• the principal form is the identity
• the product of the factors 1 + pᵃ is twice the number
• the ratio of count to prediction falls at every step
• the rectangle holds every divisor
• the reduction ends at one of the reduced forms
• the search radius is a whole number between 3 and 8
• the second tree's leaves multiply back to the number
• the sequence is long enough for the rows asked for
• the share at the end of the range is close to a quarter
• the share climbs from the first band to the last
• the share falls as the threshold rises and stays positive through four
• the share of distinct entries keeps falling
• the shortfall grows as smoothness is asked of smaller primes
• the sieve finds exactly the known multiperfect numbers below the bound
• the smallest Hilbert number with two factorisations is 441
• the smallest norms are 1, 4, 5
• the smallest odd abundant number
• the step budget is a whole number between 10 and 800
• the stretch shows the driver holding
• the table's distinct entries match the bitset count
• the three other lines are a silent prime, a composite with one shared-factor representation, and a composite with two
• the two conjugate factors have norms multiplying to 36
• the two trees start differently
• the variance trails log log N by more than one and a half
• the view is one of lattice, grid, aliquot, mersenne, failure, trajectory, driver, fates, abundancy, primitive, abundantdensity, abundancydist, primitivesum, hilbert, hilbertgroup, hilbertatoms, hilbertmulti, monoids, covering, divfail, euclidgi, motzkin, hilbertlemma, euler41, rabinowitsch, multitable, abundancybars, primesneeded, multiscan, sigmachain, solitary, friends, clubs, ten, outlaws, forms, reduction, classnumbers, formprimes, compose, classgrowth, cldivide, clbands, clgenus, cltorsion, clrank, idoneal, idforms, idprime, idresidues, idsearch, spoof, spoofsearch, eulerform, oddprimes, oddclose, omegahist, omegagrowth, omegastandard, loglog, omegaextra, omegamodel, weird70, weirdprim, weirdlist, weirdexcess, weirdodd, pracmake, pracegypt, pracrule, praccount, pracgoldbach
• there are primitive abundant numbers in the range
• three odd primes can already reach 2: 3, 5 and 7
• treating 22021 as prime, the divisor sums multiply to exactly 2N
• x² + 5y² takes exactly the primes 1 or 9 mod 20
• ρ(10) is a few times 10⁻¹¹
• σ(m) is odd exactly for squares and twice squares, up to 2,000
• ω/2 is more than 1 from every integer of the field

• Ramanujan: 5 divides p(4) ×342857
• Gordon's identity, i = 1, at n = 0 ×93
• first identity at 0 ×41
• second identity at 0 ×41
• as many partitions of 30 have largest part 1 as have 1 parts ×30
• the coefficient at 0 is what the theorem says ×27
• p(0) is 1 ×21
• the product's coefficient at 0 is p(0) ×19
• the two products agree at 0 ×19
• p(4) is divisible by 5 ×11
• log p(10) is below π√(2n/3), as it is for every n ×10
• runs of length 1 are as common as a coin's ×10
• the estimate at 10 is within a factor of two of the count ×10
• modulo 5, class 0 has its fair share of zeros ×5
• modulo 5, class 1 has excess zeros ×5
• the ranks modulo 5 split the 30 partitions of 9 into 5 equal classes ×4
• the number partitioned is a whole number between 50 and 5000 ×3
• the partitions of 8 into odd parts and into distinct parts come out equal ×3
• modulo 2 every residue is equally common ×2
• modulo 5 zero is at least as common as the congruence plus an even spread makes it ×2
• the number of coefficients shown is a whole number between 6 and 20 ×2
• the number sorted by rank leaves 4 on division by 5 ×2
• the number whose partitions are sorted by rank is a whole number between 4 and 14 ×2
• the range is a whole number between 12 and 60 ×2
• |τ(p)| ≤ 2p^(11/2)
• 232,848 plane partitions fit in the 4 × 4 × 4 box
• a partition whose parts differ by at least 2
• and at this size it is under 0.04 everywhere
• and changes the number of parts by exactly one
• and it is the largest one that fits
• and it meets the diagonal at ln 2 / c
• and the estimate is closer at the far end than at the near one
• and the partitions the move cannot touch are exactly what survives
• and the ratio climbs towards that ceiling rather than away
• at least one of the divisible values is in the table
• counting by Durfee square accounts for every partition exactly once
• Deligne's bound |τ(n)| ≤ d(n) n^(11/2)
• doing it twice gives back what it started as
• each doubling of n brings the ratio closer to ln 2 / c
• each sample is a partition of n
• every listed partition adds to the number it partitions
• half a million partition numbers are within half a percent of half even
• MacMahon's box formula matches the count by transfer
• MacMahon's product counts the plane partitions found one by one
• partitions of 10: as many with gaps of two as with parts ≡ ±1 (mod 5)
• Ramanujan's value at e^(−2π): √((5 + √5)/2) − (1 + √5)/2
• removing the staircase leaves a partition into at most k parts
• the angles fit the Sato–Tate density
• the average distance to the curve falls at each size
• the columns never increase
• the congruence drawn is the one modulo 5 or modulo 7
• the conjugate is a partition of the same number
• the continued fraction equals q^(1/5) times the ratio of the two products
• the cube's count grows like e^(c n²) with c = (9/2) log 3 − 6 log 2
• the curve is its own reflection in the diagonal
• the depth of the continued fraction is a whole number between 20 and 200
• the drawn plane partition has 29 cubes
• the even count far exceeds the square root
• the first values are 1, −24, 252, −1472, 4830, −6048, −16744, 84480
• the forced class is all zeros
• the gap to the Gumbel law shrinks as n grows
• the guess first fails at 6, where it predicts 141 and there are 140
• the largest number counted is a whole number between 20 and 120
• the largest number in the table is a whole number between 12 and 40
• the marked values are whole numbers inside the range drawn
• the move gives another partition of the same number into distinct parts
• the number being partitioned is a whole number between 3 and 12
• the number is a whole number between 5 and 14
• the number of samples is a whole number between 1 and 8
• the number whose distinct partitions are paired off is a whole number between 3 and 14
• the numbers outside the congruence class are not all divisible either
• the partition view is one the family draws
• the partitions listed are all of them
• the parts are a descending list of whole numbers
• the ratio climbs towards 1 from below
• the recurrence divides exactly
• the recursion divides exactly
• the restricted counts are not the unrestricted one
• the rows never increase
• the same count against parts ≡ ±1 mod 6 fails somewhere
• the series drawn is the partition product or Euler's identity
• the signed count of distinct-part partitions is the product's coefficient
• the square, the arm and the leg account for every dot
• the staircase 1 + 3 + 5 + … holds k² dots
• there is a Durfee square to draw
• three sizes
• turning the diagram over twice gives it back
• volume v and volume 64 − v are equally common
• τ(mn) = τ(m)τ(n) when m and n share no factor
• τ(n) ≡ n σ₃(n) mod 7
• τ(n) ≡ n σ₉(n) mod 5
• τ(n) ≡ σ₁₁(n) mod 691
• τ(n) is not zero
• τ(p) mod 23 is 2, −1 or 0 as x³ − x − 1 has three, no or one root mod p
• τ(p^(k+1)) = τ(p)τ(p^k) − p¹¹τ(p^(k−1))

• a(0) is the whole number given by the binomial sum ×61
• the denominator of b(0) divides 2·lcm³ ×61
• the denominator of b(0) divides lcm² ×61
• term 0 follows the pattern ×30
• past the threshold the tail at level 1 lies strictly between 0 and 1 ×14
• the search over divisors of 2 agrees with whether 2 is a perfect square power ×14
• the derivative of order 7 at zero is a whole number ×12
• and below 1/2 ×11
• at q = 2 the tail is positive ×11
• q! times the first 3 terms is a whole number ×11
• the degree-1 integral is positive, because the integrand is ×9
• the fraction 1/1 is no closer to √2 than 1/(3q²) allows ×9
• the largest value of the degree-1 polynomial is (π²/4)ᵏ/k! ×9
• and 4 has one such root, not several ×7
• the search over divisors of 2 agrees with whether 2 is a perfect cube power ×7
• Aₙ = q·e − p for the convergent 1/1 ×5
• Bₙ = p − q·e for the convergent 1/0 ×5
• Cₙ = p − q·e for the convergent 2/1 ×5
• the 1th truncation beats the exponent 2 ×5
• and it does so with room to spare: the error is below q^(−2) ×4
• some truncation beats the barrier of degree 2 ×4
• the search over divisors of 2 agrees with whether 2 is a perfect fourth power ×4
• at x = 0.3 each deeper convergent is closer to tan x ×3
• the fraction converges to tanh 0.5 ×3
• √2 is 1; 2, 2, 2, …
• √2: all 16 terms are certain
• a fraction's denominator is not zero
• about six in π² numbers are squarefree
• and (1 + tanh ½)/(1 − tanh ½) is e
• and ends below one millionth
• and is below twice the peak, since sin integrates to two over the interval
• and it is closer than 1/q², which is what makes the exponent exactly two
• Apéry's fractions approximate only like one over the denominator to a power a little above one
• at least one of the numbers drawn has an irrational root, or the figure argues nothing
• between one and three increasing barrier degrees, none above the number of truncations drawn
• between two and five increasing degrees, each between 1 and 12
• between two and ten whole numbers, each between 2 and 64
• e follows its pattern
• e or its square root
• e: all 16 terms are certain
• every convergent is within one over its denominator squared
• for √2, q² times the error never falls below a third
• for e the lowest values drift down, slowly
• golden ratio: all 16 terms are certain
• its denominator is a whole number between 1 and 200
• its roots are all different
• multiplying the sixteen numbers gives that whole number
• q times the error falls at every step
• so it is strictly between 0 and 1, where no whole number is
• sums of more roots can come closer, faster
• the bounds fix at least as many terms as are drawn
• the constant is ζ(2) = π²/6
• the constant is ζ(3)
• the denominator is a whole number between 1 and 5
• the error shrinks like (√2 − 1)⁴ⁿ
• the error shrinks like φ⁻⁵ⁿ
• the fraction is at least close enough to π for the picture to be about π
• the golden ratio is all ones
• the largest denominator tried is a whole number between 3 and 14
• the lcm grows slowly enough for both proofs
• the number of convergents drawn is a whole number between 4 and 9
• the number of tail terms summed exactly is a whole number between 12 and 60
• the number of truncations of the constructed number is a whole number between 3 and 5
• the numerator is a whole number between 1 and 5
• the numerator of the fraction π is supposed to be is a whole number between 2 and 400
• the polynomial has degree two to the number of roots
• the product of the whole-number multiplier and the error falls to nought
• the product over all sign patterns is a non-zero whole number
• the range drawn reaches a degree whose bound is below one, which is the whole argument
• the root taken is a whole number between 2 and 4
• the smallest gap is far below the average
• the tail falls as the denominator grows
• the two sums differ, by less than a hundred-millionth
• the view is one the family draws
• there are derivatives to show
• twelve levels at x = 1 give tan 1 to double precision
• ζ(2) or ζ(3)
• π: all 16 terms are certain
• π's fifth term is 292
• ψ(n) is the logarithm of the lcm

• the count on the circle of squared radius 1 is 4(d₁ − d₃) ×64
• 3, 4, 5 is a Pythagorean triple ×10
• the points on the circle of radius √25 are 4(d₁ − d₃) ×8
• 5 = 2² + 1² ×7
• the tetrahedron of height 1 has no lattice point inside it ×7
• the area is 20 interior points plus half of 7 on the edge, less one ×5
• the count at dilation 1 is the Ehrhart polynomial's value ×4
• the count at k = 1 matches the cubic fitted from the first value ×4
• n is a whole number between 2 and 70 ×2
• the number of columns in the grid is a whole number between 4 and 8 ×2
• the range is a whole number between 10000 and 2000000 ×2
• 1 has solutions both on and off its family
• 23 and 239 need nine, 454 eight, 8,042 seven
• 23 has one way with nine cubes, 239 two, and 454 two with eight
• 31·16ᵐ needs sixteen fourth powers
• a group with entries only up to two fails to identify two of the classes
• a nine-degree sector holds its share to within ten per cent at every bound
• a number congruent to r modulo 16 never needs fewer than r fourth powers
• a polygon's boundary count and its dual's add to twelve
• a prime is a sum of two squares exactly when it is one more than a multiple of four
• a square root of −1 exists modulo a prime one more than a multiple of four
• a² + b² lies at most 2b past n
• and a self-dual one has six boundary points, which is half of twelve
• and dividing by the radius itself tames it further still
• and every basis along the way spans a cell of the same area
• and exactly four on it, which are its own corners
• and from entries up to three onward the count stops moving
• and is one more than a multiple of four
• and it is the first one below it — the row above is still above
• and it really is in the lattice
• and Scott's inequality caps the boundary count at nine
• and so within 2√2 · n^(1/4) + 1
• and some attain the bound
• and stays inside three times the two-thirds power throughout
• and stays inside twice the cube root over the range drawn
• and stopping one step earlier would not have given two squares
• and the answer is one of the points the circle of that radius actually passes through
• and the candidate really does square to −1
• and the density is falling, which is the statement the constant qualifies
• and the disc of radius two catches thirteen
• and the flat ones are excluded rather than ignored
• and the point it came from is on the unit circle
• and the rest pair off
• and they have between three and six corners
• and twelve has six
• at every bound the primes' angles are more even than independent draws
• at least two heights are drawn
• at width X^(−0.6) there are far more empty sectors than chance allows
• between one and five powers are drawn
• both polynomial families add up for every t
• cubes leave only 0, 1 or 8 on division by 9
• D is a whole number between 2 and 40 and not a perfect square
• D is between 2 and 40
• D is not a perfect square, or the equation has only the trivial solution
• each of these walks stops inside its disc
• each recorded representation adds up exactly
• each reflection strictly lowers the denominator
• every cosine averages below one per cent at two million
• every edge lies at lattice distance one from the interior point
• every lattice triangle in the grid satisfies the identity
• every marked point really is on the circle
• every one of the 2300 triples of grid points was reached
• every one-degree band holds its share to within four per cent
• every power is again a solution
• every recorded triple adds up
• every solution for 2 in the box lies on its family
• every solution up to the largest drawn is a power of the smallest
• every step length up to 112 is stopped by a gap below two million
• exactly seven numbers need nineteen fourth powers
• for some height the linear coefficient is negative, so it counts nothing
• for x² + 3y² the centre sits at exactly 1
• four consecutive sums of two squares never happen
• four of the sixteen are their own dual
• in five dimensions the typical error is a fraction of one shell's population
• Jacobi's four-square formula
• most empty sectors lie beside slopes whose two terms are both odd
• n names one circle and is read only by the default view; the other views take their own parameters
• no gap exceeds the elementary bound
• no number 3 more than a multiple of 4 is a sum of two squares
• no number leaving 4 or 5 on division by 9 is reached
• no polygon with an interior point has more boundary points than Scott allows
• no two slopes give the same triple
• one extra for the hole repairs it, which is the Euler characteristic in disguise
• one has one divisor
• one of them attains Scott's bound
• one to three numbers up to 100,000
• one to three rational turns
• p names the prime the two squares are produced for and is read only by the cornacchia and reduce views
• Pick's formula with one interior point makes twice the area the boundary count
• Pick's theorem holds for every polygon swept
• primes a² + 1 found two ways
• seventeen numbers up to 600 need eight or nine cubes
• some circles miss the lattice entirely, which is the whole question
• sums of three squares miss exactly the numbers 4ᵃ(8b + 7)
• the answer satisfies the congruence the chain started from
• the average count grows by between one and two per tenfold enlargement
• the average count of ordered four-cube sums matches the volume prediction
• the base point is on the circle
• the bound is attained only at one interior point
• the bound the count runs to is a whole number between 2000 and 200000
• the box misses exactly the eight numbers the large searches were needed for
• the box of side 5,000 reaches between 85 and 92 per cent
• the box the polygons are drawn from is a whole number between 3 and 4
• the centre of a square cell is at squared distance 1/2 from the corners
• the chain ends on a lattice point
• the circle of squared radius one carries four points
• the count times √(log x) over x is near Landau's constant at the top of the range
• the counts dip beside the slope 1/3
• the disc of radius one catches five points
• the divisors of everything up to six add to fourteen
• the dual of one of the sixteen is another of the sixteen
• the error averages out to nought
• the error divided by the cube root is the tamer ratio
• the error divided by the two-thirds power is the tamer ratio
• the error is not bounded by twice the square root over this range
• the factorisation test and the circle agree about every number up to two hundred
• the first remainder below the square root, with its partner, squares and adds to the prime
• the identity as stated gives the wrong area for a region with a hole
• the largest bound counted is a whole number between 200 and 4000
• the largest gap shrinks from the smallest bound
• the largest prime to test is a whole number between 10 and 200
• the largest radius counted is a whole number between 40 and 400
• the largest squared radius drawn is a whole number between 12 and 64
• the last number is a whole number between 59 and 99999
• the last numbers needing nine, eight, seven and six cubes
• the last numbers needing seventeen and eighteen are 13,792 and 1,248
• the lattice half-width is a whole number between 1 and 16
• the leading coefficient is the volume for every height drawn
• the measured growth of the largest error matches the known orders
• the new point is on the same circle
• the number is a whole number between 1 and 200
• the number is prime
• the number of cells is a whole number between 60 and 400
• the option is one this view reads
• the parts add up to the number
• the polygon has between 3 and 16 whole-numbered corners and some area
• the polygon is one of blob, triangle, comb, thin
• the prime is one more than a multiple of four
• the prime test and a direct search agree
• the prime the lattice is built from is a whole number between 5 and 200
• the prime the two squares are found for is a whole number between 5 and 4001
• the remainder after the largest square is at most 2a
• the second differences are twice the area
• the share needing six vanishes and the share needing five is falling
• the share of Gaussian integers near radius r that are prime is about 4/(π ln r²)
• the shortest vector of the lattice has length squared exactly the prime
• the slices add up to the count from sums of three squares
• the slopes are proper positive fractions
• the solution found satisfies the equation
• the starting point is on the circle
• the sweep found a substantial family
• the sweep grid is between 3 and 6 points across
• the three-square count equals Gauss's class-number formula
• the view is one the family draws
• the volumes differ while the counts do not
• there are sixteen classes to pair up
• there are sixteen lattice polygons with a single interior point
• three cubes reach every remainder but 4 and 5
• twice the area is a whole number, which is why halves are the only fractions in the formula
• two numbers need nine, fifteen need eight, 121 need seven
• upTo names a range and is read only by the gaussian, count and density views
• while a polygon with no interior point can exceed the bound, which is why it needs one
• wide sectors fluctuate far less than chance and thin ones nearly as much
• with steps of √10 the farthest prime reached is at distance 1,024.4
• with steps of √18 the walk reaches the edge of a disc of radius 4,500
• with steps of length at most 2 the walk from 1 + i is trapped, its farthest prime 17 + 42i

• ?(1/11) is the binary fraction in the same position ×1023
• 0/1 and 1/13 have determinant one and nothing simpler between them ×359
• 3/2 is in lowest terms ×108
• the circles on 0/1 and 1/7 touch ×56
• s(1) and s(2) are coprime ×48
• the 1th ratio's numerator agrees with the Calkin–Wilf walk ×48
• the ratio 1/1 appears once ×48
• 0/1 and 1/7 are Farey neighbours ×24
• 1/1 arrives already in lowest terms ×15
• and its mediant 1/1 is already in lowest terms ×13
• s(2) counts the hyperbinary representations of 1 ×12
• the 1th arc crossed is the descent's 1th interval ×10
• run 1 of the turns is the continued fraction's quotient ×5
• the identity for n ≡ 3 (mod 4) gives 4/n ×3
• the sequence has one entry per coprime pair up to 7 ×2
• ? of the expansion of all ones is 2/3
• ? of the expansion of all twos is 2/5
• ?(1 − x) = 1 − ?(x)
• ?(x/(1 + x)) = ?(x)/2
• |M(x)| stays below √x throughout the range drawn
• √2 − 1 has every quotient two
• 4/n is a sum of three unit fractions for every n drawn
• above 100, the fewest ways belong to a prime that leaves 1 on division by 24
• and at most its right bound
• and is closer to the target
• and it is within a per cent of 3n²/π² by the end
• and its denominator does too
• and so do the three fractions found by search
• and the path closes on the target
• and their lengths add to one
• circles that are not neighbours stay clear of each other
• depth d cuts the interval into 2^d pieces
• each path is a word of at most eight L's and R's
• each record has a larger denominator than the last
• every denominator up to 3,000 has a fraction with quotients at most 5
• every prime ≡ 1 (mod 24) up to the limit has a three-term decomposition
• every prime has at least one good fraction and the share is a share
• how many terms are drawn is a whole number between 8 and 64
• how many turns are taken is a whole number between 6 and 34
• M(10) = 1 − 1 − 1 + 0 − 1 + 1 − 1 + 0 + 0 + 1 = −1
• n leaves remainder 1 on division by 4 and is small enough to draw
• n times the sum of squared misfits stays below one across the orders drawn
• neighbouring Farey fractions have cross-difference one
• neighbouring Farey fractions have determinant one
• neighbouring nodes have determinant one
• no denominator up to 100,000 lacks a fraction with quotients at most 5
• no fraction is produced twice
• no two words produce the same matrix, so the tree never rejoins
• only the class of 1 mod 24 escapes the three identities
• reading the drawing left to right reads the fractions in increasing order
• the chain from 4/n with n ≡ 1 (mod 4) has four steps
• the count is the length of the generated sequence
• the counting exponent approaches twice the dimension
• the denominator has a fraction with quotients at most 5
• the denominator is a whole number from 11 to 100,000
• the depth is a whole number between 2 and 5
• the depth of the tree checked is a whole number between 3 and 14
• the descent passes at least three fractions within 1/q² of the target
• the descent produces several record approximations
• the drawing scale is a whole number between 200 and 400
• the endpoint each crossed arc keeps spells the descent's word
• the Farey fractions, placed on the circle and added, give the Mertens function
• the Farey order is between 3 and 6
• the four gaps add up to 4/n
• the function never decreases
• the largest denominator drawn is a whole number between 5 and 30
• the length carrying half the rise falls with depth
• the line crosses exactly as many arcs as the descent has intervals
• the matrix for "L" has determinant ±1
• the matrix for "LR" has determinant ±1
• the matrix for "LRL" has determinant ±1
• the matrix for "LRLR" has determinant ±1
• the matrix for "LRLRL" has determinant ±1
• the matrix for "R" has determinant ±1
• the matrix for "RL" has determinant ±1
• the matrix for "RLR" has determinant ±1
• the matrix for "RLRR" has determinant ±1
• the matrix for "RR" has determinant ±1
• the matrix for "RRL" has determinant ±1
• the matrix for "RRLR" has determinant ±1
• the matrix for "the root" has determinant ±1
• the mediant is at least its left bound
• the mode of the mediant family is one of tree, farey, ford, matrices, diatomic, approx, tessellation, questionmark, singular, fareydisc, franel, landau, mertens, expsum, fareycount, estree, esneighbours, eschains, escount, esclasses, eshard
• the number is one of phi, root2, pi, e
• the order is a whole number between 2 and 12
• the path fits the drawing
• the search's largest fraction is the tree's first gap
• the target is one of phi, root2, pi, e
• the total misfit grows like the square root of the order
• the tree has 2^D − 1 nodes to depth D
• the tree is full to its stated depth
• the tree takes two steps when n ≡ 3 (mod 4) and four when n ≡ 1
• the typical count grows like q to the power 2δ − 1
• the window is a sub-interval of the unit interval
• three unit fractions suffice for every n drawn
• two to six increasing depths, up to 18
• up to which index the hyperbinary count is checked is a whole number between 4 and 24
• with nothing left over in the other direction
• with quotients at most 4 only 6, 54 and 150 fail up to 100,000
• φ − 1 has every quotient one

• Fermat's theorem holds at 3 ×549
• the largest order mod 3 is λ(3) ×58
• the primitive roots of 3 number φ(p − 1) ×29
• the search and the rule agree at 2 ×29
• 1 to the power 4 is 1 modulo 15 ×28
• 1 to the power 13 − 1 is 1, which is Fermat's theorem ×12
• the order of 1 divides 13 − 1 ×12
• exactly φ(1) residues have order 1 ×8
• the class of numbers sharing n/1 with n has φ(1) members ×6
• the order of 2 divides λ(561) ×6
• the order of 2 falls short of 561 − 1 ×6
• the Wieferich primes to base 2 below 200000 ×6
• base-2 pseudoprimes up to 1000 ×5
• Carmichael numbers up to 1000 ×5
• the modulus is a whole number between 5 and 31 ×4
• the powers of 2 run through every residue ×4
• 1009 has a primitive root ×3
• 11 to the 1008 is 1 mod 1009 ×3
• Chernick's construction at k = 1 satisfies Korselt's divisibility ×2
• no proper divisor of 4 works for every unit ×2
• the base is a whole number between 2 and 12 ×2
• the bound is a whole number between 12 and 42 ×2
• the length is a whole number between 3 and 7 ×2
• the number is a whole number between 6 and 24 ×2
• the number of black beads is a whole number between 1 and 6 ×2
• the number of colours is a whole number between 2 and 4 ×2
• (x + 1)ⁿ ≡ xⁿ + 1 modulo n exactly when n is prime
• 3,215,031,751 = 151 · 751 · 28,351 passes the bases 2, 3, 5 and 7 and fails 11
• 5 is a fifth power for about one prime in twenty
• a base passing every check exists exactly when the number is prime
• a Carmichael number has at least three prime factors
• a constant fitted at a tenth of the range predicts the full count within fifteen per cent
• a few numbers between 100 and 3000
• a large share of odd composites have only the two trivial liars
• an odd composite from 100 to 2,000
• and exactly a of them are single
• and no smaller power along a prime is
• and reach each of them once
• and the largest order divides φ(n)
• and the number of non-constant classes is (a^p − a)/p
• Artin's constant, as a product over the primes
• at least one composite below 2000 survives every base
• at least one product was checked against every base rather than by the criterion alone
• at least one residue has order p − 1, which is what being cyclic means
• at least two composites below 2000 survive base 2
• at least two values of k below 12 give three primes
• below ten thousand there are five strong pseudoprimes to base 2, and below ten million fewer strong than ordinary ones
• both verdicts occur inside the bound, so the table is deciding something
• Chernick's product satisfies Korselt's criterion
• each base's share is near its prediction
• every arrangement of the black beads is drawn once
• every base coprime to 1729 returns 1
• every Carmichael number is a base-2 pseudoprime
• every class is a single string or a full ring of p
• every middle entry of row n is a multiple of n exactly when n is prime
• every order divides the largest one
• every other class holds exactly p strings
• every ring's size divides the length
• every string of the given length is drawn
• index one — 2 a primitive root — has Artin's share
• Mertens's estimate matches the exact sum of 1/p at a million
• Monier's formula counts the strong liars exactly
• more than one order occurs, so the picture is separating the residues
• no odd composite other than 9 below 100,000 has more than a quarter of its bases lying
• not every residue is a generator, so the list is a selection
• one to three bases
• only 1093 and 3511 keep their order
• p divides a^p − a, which is what the count just showed
• some of them are caught by another base, so the two columns differ
• the base is coprime to the modulus
• the classes account for every number from 1 to n
• the classes account for every string
• the classes of size one are exactly the constant strings
• the count and the rings agree
• the criterion and the exhaustive base test give the same verdict
• the drawn ring has as many beads as the base's order
• the figure draws every string, so there is a ceiling on how many
• the first Carmichael numbers are 561, 1105, 1729
• the first Chernick numbers come from k = 1 (1729) and k = 6
• the largest k searched is a whole number between 1 and 40
• the last row is a whole number between 10 and 26
• the length is prime
• the length is prime — the whole argument needs it to be
• the measured exponent creeps upwards
• the measured share settles near the predicted density
• the modulus is prime
• the modulus is prime, so every non-zero residue has an order
• the number has at least three divisors, so the identity is not trivial here
• the number is a prime below a million
• the number is a whole number below a million
• the number is composite — the criterion is about composites
• the number of beads is a whole number between 4 and 9
• the number tested is a whole number between 9 and 400
• the only zeros are 1093 and 3511
• the orbit closes
• the orders account for every residue
• the quotients are computed in exact whole numbers below 9,000
• the range of primes is a whole number between 100000 and 3000000
• the reduced polynomial test with a = 1 separates these primes from these composites
• the searches find 2,047, 1,373,653 and 25,326,001
• the strong liars counted base by base match Monier's formula
• the units are counted, and there are φ(n) of them
• the view is one the family draws
• there are φ(13 − 1) generators
• there is more than one base to test
• when the length is prime, no arrangement with some black and some white is carried to itself by a rotation
• whenever 5 is a fifth power modulo p, it is also a square modulo p
• λ(n) = n − 1 exactly when n is prime

• two primes agree modulo 3 and disagree about x² + 27y² ×158
• 3 is represented by x² + 1y² exactly when 3 mod 4 is one of the classes ×134
• every step is a whole unit, because 1 is not a multiple of 13 ×70
• the sign of multiplication by 3 is (3 | 11) ×22
• (−1 | 3) is decided by 3 mod 4 ×17
• (2 | 3) is decided by 3 mod 8 ×17
• Gauss's lemma counts 1 folds for −1, and its parity is the symbol ×17
• Gauss's lemma counts 1 folds for 2, and its parity is the symbol ×14
• the modulus deciding x² + 1y² is at most 4n ×3
• the modulus is a whole number between 5 and 31 ×3
• 3 folds gives the same answer as Euler's criterion ×2
• (p | q) comes out of the count above it
• (q | p) comes out of the count below the line
• a prime with both symbols positive is 1 modulo 8, which is the two conditions met at once
• and every one of them lands in the bottom half
• and exactly half the multipliers are squares
• and it has no imaginary part
• and points up, which is the part Gauss took four years over
• and positive, which is the part Gauss took four years over
• and their product is minus one to the power of the whole rectangle
• between two and five forms are drawn
• both are odd primes and they are different
• each form's coefficient is a whole number between 1 and 27
• every moved cycle has the same length, which is the order of the multiplier
• every number from 1 to 42 is a square mod 366,791
• every point is counted exactly once, on one side or the other
• every record lies below 2(ln p)², the bound the Riemann hypothesis would give
• for a prime 1 modulo 4 the sum is real
• for a prime 3 modulo 4 the sum is purely imaginary
• no lattice point lies on the diagonal, because the primes are coprime
• the average approaches Erdős's constant from below
• the bound the congruence check runs to is a whole number between 2000 and 40000
• the cycle count and the inversion count give the same sign
• the drawing holds both kinds of form, which is what it is for
• the first prime is a whole number between 3 and 31
• the first prime whose least non-residue is 43 is 366,791
• the folded values are all different
• the folds for 2 are the k above p/4
• the largest modulus tried is a whole number between 20 and 400
• the largest prime drawn is a whole number between 60 and 400
• the largest prime in the table is a whole number between 11 and 97
• the least non-residue is always a prime
• the length of the sum, squared, is the modulus
• the modulus is an odd prime
• the multiplier is a whole number between 2 and 30
• the multiplier is not a multiple of the modulus
• the points below the line are the sum of the floors of kq/p
• the remainder mod 24 decides whether n(p) is 2, 3 or larger
• the second prime is a whole number between 3 and 31
• the share with least non-residue the k-th prime is close to 2⁻ᵏ
• the square of the sum is plus or minus the modulus
• the view is one the family draws
• the walk never strays more than √p ln p from nought

• the class 0 shares a factor with 4, so it can hold at most the factor itself ×20
• 211 leaves remainder 1 on division by 2 ×10
• the narrowest admissible 2-tuple has width 2 ×9
• the limit is a whole number between 10 and 400 ×5
• 2 classes share no factor with 4 ×4
• the staircase ends at π(3000) ×4
• every listed prime is already in the class 3 mod 4 ×2
• so some prime factor of it is 3 mod 4 — the classes multiply, and a product of ones is one ×2
• the class 3 leads at almost every bound ×2
• the modulus is a whole number between 3 and 14 ×2
• the number built is 3 mod 4 ×2
• (0, 2, 4) is not admissible: one of three numbers two apart is a multiple of 3
• a Mersenne prime has a prime exponent
• a narrowest admissible pattern is found
• a pattern is admissible exactly when it leaves a remainder free mod every prime
• a small prime works for every even number in range
• a square left unstruck holds a prime, and a struck one does not
• a truncated constant recovers more primes as it carries more digits
• after small cases the first odd number that is not a prime plus a power of two is 127
• after the fifth term the largest-factor sequence never returns below fifty thousand in the terms computed
• an earlier term leaves remainder 1
• an even truncation is an upper bound
• and 168 below a thousand
• and an odd one a lower bound
• and every combination of exponents appears once
• and it tracks the logarithm of a logarithm to within a twelfth throughout
• and modulo three it never falls behind over this range
• and over the last decade of the range the diverging sum gains more than twice as much
• and sifting the odd half leaves exactly the primes above √x
• and some of the lower bounds are negative, which is no bound at all
• and the first bound at which it does not is 26,861
• and the number of them in the interval is never far below n / ln n
• and the one prime it holds divides the modulus
• and the ratio to x / ln x comes down as x grows
• and the twin pairs track theirs
• and there are almost none of them in this range
• and they hold them in near-equal numbers, which is more than Dirichlet's theorem claims
• and thirty is favoured over its neighbours
• between two and five odd primes are listed
• both classes hold primes below the bound
• both ends are prime
• Chen's primes greatly outnumber the twins
• each is near the share an even split would give
• each prime in the chain fixes about three times as many digits of A as the last
• each prime lies below the next cube
• every class coprime to the modulus holds primes
• every count is within fifteen per cent of its prediction
• every exponent has at least one prime dividing it
• every interval from n to 2n holds a prime
• every number strictly between is composite
• every pair count is within eight per cent of Hardy and Littlewood's prediction
• every prime below the bound lands in exactly one class
• every prime factor of one more than a square of an even number is 1 mod 4
• every prime factor of the constructed number is outside the list
• every record so far sits below (log p)² with room
• every window between consecutive powers holds a prime
• gaps of six outnumber gaps of two and four in this range
• Ingham's exponent already allows the power 3, and the best known allows any power above 2.106
• Mertens' product is within two per cent of e^γ ln p at 1,100
• multiplying the factors out gives exactly the sum over the numbers they build
• no listed prime divides it
• no record below thirty million reaches (log p)²
• one gap between each consecutive pair
• seventy-one composites in a row
• sifting the even half by every prime below √x leaves nothing
• the bound the postulate is checked to is a whole number between 100 and 50000
• the chain agrees with the published one
• the column count is a whole number between 4 and 25
• the constant agrees with the published value
• the construction is the one that subtracts or the one that squares
• the count runs above x / ln x over this range
• the coverings drawn are those of 78557 and 271129
• the factorisation multiplies back to the number built
• the factors are between 2 and 4 primes
• the five classes cover every whole number
• the full inclusion-exclusion agrees with the count made by sifting
• the gap of 210 first appears after 20,831,323
• the gap of 72 first appears after 31,397
• the gaps add up to the distance from 2 to the last prime
• the grid of whole numbers drawn is a whole number between 24 and 144
• the largest bound counted is a whole number between 5000 and 200000
• the largest number sieved is a whole number between 100000 and 2000000
• the largest prime sifted by is a whole number between 11 and 43
• the list is between two and six primes
• the logarithmic integral runs above the count at every checkpoint drawn
• the marked checkpoints are whole numbers inside the range
• the n up to 170 with n! + 1 prime are the known list
• the n up to 170 with n! − 1 prime are the known list
• the next term divides the number
• the number built stays inside exact arithmetic
• the number of intervals drawn is a whole number between 6 and 40
• the numbers marked are exactly those the product builds
• the option is one this view reads
• the partial sums swing far past the answer before settling
• the pass number is between 0 and 8
• the power each factor is truncated at is a whole number between 2 and 5
• the primes 1 mod 4 are about half of all of them
• the primes one more than a square track the conjectured count
• the primes p up to 1,100 with p# + 1 prime are the known list
• the primes p up to 1,100 with p# − 1 prime are the known list
• the race is drawn modulo three or four
• the range sifted is a whole number between 10000 and 1000000
• the range the sum over primes is drawn to is a whole number between 1000 and 1000000
• the residue method's factor divides p# + 1
• the sieve finds the 1,857,859 primes below thirty million
• the sieve found every prime below the bound
• the smallest factor of p# + 1 exceeds p
• the smallest factor of p# − 1 exceeds p
• the squares left standing below 100 are the primes
• the sum of the reciprocals of the primes keeps growing
• the sum over all primes has passed 2.8 and is still climbing
• the test finds the known exponents for 3·2ⁿ + 1
• the two halves' shares of multiples of every d agree to within a fifth
• the view is one the family draws
• there are 25 primes below 100
• up to this bound the sum never rises above zero after n = 1
• while the ratio to the logarithmic integral is nearer one by more than a factor of four
• while the sum over the twins is still under 1.8
• π(x) stays above x / ln x throughout this range

• the map has an orbit of least period 1 ×6
• above the critical parameter the trajectory visits both lobes
• and at the finest it is a few per cent
• and each is narrower by the stretch factor
• and even at the cusp, where a straight line cannot fit, the departure stays small
• and it is the Lorenz system's own exponent, near nine tenths
• and never rises above it again
• and no window it finds contains 1.4
• and settles on neither fixed point
• and the strips are laid out in order without overlapping
• and there are parameters nearby where it is negative — periodic windows
• areas shrink at every step
• at a = 1.4 the exponent is about 0.42
• at every finer grid, 90% of the time fits in a smaller share of the area
• below the critical parameter it spirals into a fixed point
• between two and four starting gaps, each a whole power of ten from −2 to −12
• both orbits are certified to exist and to attract
• each cycle returns to its start
• each image has 0.3 times the area of the one before
• each run passes many peaks
• enough returns to see a shape
• every box's interval image lies inside the set of boxes
• every parameter sampled in the proved region has a positive exponent
• every periodic point found is a saddle
• every point of the boundary lands strictly inside the quadrilateral
• every surviving strip lies inside the square
• every window still holds some of the orbit
• half the time is spent in under a quarter of the cells the orbit visits
• Krawczyk's test certifies a stable periodic orbit
• Newton's method converges to the orbit
• no box's image leaves the grid
• no periodic orbit's average is the typical one
• orbits from different starts spend their time the same way
• scans close to 1.4 at ever finer spacing find fewer windows, and the finest finds none
• so the total width falls, and the limit has none
• the closed-form fixed points really are fixed
• the contraction is between 0.1 and 0.4
• the direction never leaves the cone and every step stretches by at least 1.4
• the first image lies inside the region, so every image lies inside the one before
• the fixed point maps to itself
• the geometric parameter is between 0.1 and 8
• the growing set of boxes closes up
• the growth rate does not depend on how close the two starts were
• the Hénon parameter is between 1 and 1.45
• the Jacobian determinant is the contraction
• the log of the volume falls at the rate σ + 1 + β
• the long averages agree
• the map's maximum is in its interior, so it folds
• the nearest window to 1.4 the scan finds has period 19, at 1.40016
• the number of magnifications is a whole number between 1 and 4
• the number of stages is a whole number between 2 and 5
• the number of steps is a whole number between 500 and 40000
• the option is one this view reads
• the orbit at the fixed point averages something else
• the orbit stays bounded, which is what makes it an attractor
• the Prandtl parameter is between 0.1 and 40
• the Rayleigh parameter is between 0.1 and 100
• the return map is a curve, not a cloud — its typical thickness is under one per cent of its range
• the scan finds the stable orbit at the stated parameter
• the step size is between 0.0005 and 0.05
• the stretching factor is between 2.2 and 5
• the strips double at every stage
• the surviving set has a dimension between nought and one
• the trace runs from +1 at the saddle-node to −1 at the period doubling
• the trajectory gets below the bound
• the trajectory stays in a bounded region
• the two starts give the same distribution of peaks
• the windows of period ten or less are typically more than ten times wider than those of period nineteen or more
• wherever V is above the ellipsoid's bound it is falling

• the word-parallel run agrees with the rule at step 1 ×200
• running the same rule backwards returns the starting row exactly, cell 0 ×101
• at step 1 the traffic rule and rule 184 agree in every cell ×69
• after one lap the flux at density 0.025 is min(ρ, 1 − ρ) ×40
• rule 90 row 0 matches Pascal's triangle mod 2 ×40
• the ring of 3 returns after its period ×28
• with no dawdling the flow at density 0.02 is min(5ρ, 1 − ρ) ×24
• blocks of 1 are as even as a coin's would be ×12
• every block of 1 appears ×12
• every configuration of a 5-cell ring ends as its majority ×6
• a density and a dawdling chance between 0 and 1
• a density strictly between 0 and 1
• a first-order rule can only lose rows, never gain them
• above half density at least 2k − n cars are always stopped
• adding the identity to the full pile and stabilising gives the full pile back
• after two hundred thousand steps the share of 1s is within three coin-toss standard deviations of a half
• an odd ring of 7 to 13 cells
• an odd ring, so there is always a majority
• and freezes with both values still present
• and the drawn ring ends as its majority
• at this density and dawdling some cars come to a halt
• below half density every jam is gone after at most one lap
• below one half the stopped share reaches zero
• between ten and four hundred topplings the avalanche sizes fall off as a power, with exponent near minus one
• blocks or blinkers are the commonest ash
• each cycle moves the ant two cells diagonally
• every cell fires at the same step
• every cell is on or off and nothing else
• every diagonal within 24 cells of an edge settles into a period within eight thousand steps
• every random start reached a highway within 300,000 steps
• every right-hand period is a power of two
• exactly five of the 256 rules conserve the number of 1s on every small ring
• far from one half it is always right
• from an empty grid the highway begins after about ten thousand steps
• more dawdling lowers the greatest flow the road carries
• near one half it is often wrong
• no elementary rule classifies every ring
• no grain reaches the edge: the pile keeps all 65,536
• no new state appears for longer lines
• no two cars ever share a cell
• on the highway each 104-step cycle adds twelve black cells
• rule 184 never creates or destroys a car
• running the rule backwards returns the empty grid, the ant to the start, facing as it began
• stopped cars never fall below (2k − n)/n
• the acorn settles at generation 5206 with 633 cells
• the ash fills between one and six per cent of the torus
• the backward run retraces the forward count step for step
• the dawdling traffic starts free and stops only later
• the diehard vanishes after 130 generations
• the drawn table reads back as the rule number
• the drawn triangle and the measured column agree
• the identity is not the empty pile: it holds grains everywhere except in a few regions
• the left-hand periods stay at four or less
• the line fires within a bounded time
• the local vote freezes within a few steps
• the number of rows is a whole number between 4 and 160
• the option is one this view reads
• the pile holds a little over two grains a cell
• the pile stays inside the grid
• the R-pentomino settles at generation 1103 with 116 cells
• the recurrent piles are exactly as many as the spanning trees
• the right-hand periods never shrink going inwards
• the ring's length is a whole number between 20 and 240
• the rule is a whole number between 0 and 255
• the rule number is a whole number between 0 and 255
• the second-order map sends different pairs of rows to different pairs
• the seed is single or random
• the start is random or one block of cars
• the steps drawn is a whole number between 10 and 160
• the view is one the family draws
• toppling in two different orders gives the same final pile and the same number of topplings at every cell
• two to a hundred and twenty cells fit the picture
• very dense starts leave less ash than middling ones
• with no dawdling, evenly spaced cars never stop

• root 1 is reached from at least one of the starting points ×5
• a point beside root 1 runs to root 1 ×3
• the iteration cap is a whole number between 8 and 200 ×3
• the sample count is a whole number between 60 and 700 ×3
• at a root of multiplicity 2 each step multiplies the error by (m − 1)/m ×2
• the 5 roots are all distinct and all found ×2
• the span is between 0.02 and 6 ×2
• a grid of 80 to 240 starting pairs a side
• a parameter just outside the copy sends 0 to a root
• a region of starting points never reaches a root
• a secant step of 0.01 nearly always lands where Newton does, and a step of 1 much less often
• along the line, the copy's period is twice the Mandelbrot set's
• and the cycle attracts, because the product of the map's derivative round it is under one
• and the disc was sampled at 144 points
• at least one root was found
• at the centre, 0 goes to a second point and straight back
• cap is read only by the speed and compare views
• chords go flat at the first, second and third steps somewhere in the square
• doubling the step at a double root restores the doubling of digits
• each cotangent returns after n steps
• each error divided by the product of the two before tends to f″/2f′ at the root
• each marked parameter traps the critical point in an attracting cycle
• each root cubes to one
• each step of the orbit, seen through the change of variable, is a squaring
• every point called a root really is one
• every point of a small disc round the cycle reaches no root
• every root marked is a root
• every start off the dividing line runs to the root on its side
• Halley at a simple root roughly triples them
• Halley's step is Newton's step for p/√p′
• just inside, 0 falls into a cycle of period 2
• method is read only by the speed view
• Newton at a simple root roughly doubles the digits each step
• outside the copy, 0 lingers near the ghost of the cycle before reaching a root
• poly is read by the cycle, universal, speed and compare views
• so the failing set has area rather than being a boundary
• the centre found returns 0 to 0 in the stated number of steps
• the digits gained grow by the golden ratio at each step
• the histogram follows the Cauchy density
• the method is Newton's, Halley's or the relaxed Newton
• the orbit's angle doubles at each step
• the parametrised curve lies on x₀² + x₀x₁ + x₁² = 1
• the period of the cycle at the centre of the copy's largest bulb
• the polynomial is one the family knows
• the roots found are counted
• the set has as many points as the construction asks for
• the two points really do map to each other
• the view is one the family draws
• with no imaginary part left over

• the formula counts the primitive two-colour necklaces of length 1, a necklace and its colour-swap as one ×14
• the same formula counts the irreducible polynomials of degree 1 over the two-element field whose roots add to one ×12
• the logistic map has exactly as many superstable parameters of period 1 as the formula says ×9
• the 2-to-1 merging is inside its bracket ×4
• the orbit at r = 2.8 has period 1 ×4
• the lower parameter is between 0 and 4 ×2
• the number of images is a whole number between 2 and 10 ×2
• the parameter is between 3.57 and 4 ×2
• the roots of exact period 6 are as many as the formula says ×2
• the upper parameter is between 0 and 4 ×2
• a family of starting points stays bounded for 600 steps
• and so do the gaps of the cascade inside the window
• and so does the sine map
• and the windows come in the same order of periods
• at least three small distances below the window
• at the critical parameter f³ touches the diagonal
• below the critical parameter the channel is open
• chaos, then period 3, then period 6
• each mean is taken over at least a hundred quiet phases
• each merging comes at a lower parameter than the one before
• one, two, four peaks before repeating, then none
• period 1 at c = 2.5, 2 at 3.5, 4 at 4, and a period-3 window at 5.3
• rising to one side of the top, falling to the other
• the box holds a diagram and not an empty strip
• the box is a genuine rectangle inside the diagram
• the column count is a whole number between 40 and 1400
• the diagram is drawn as marks rather than as a bitmap of elements
• the distance below the window is small and positive
• the doubled orbit is found next to the old one
• the doubling points come in order
• the doublings come in order
• the gap ratio approaches Feigenbaum's constant from this side too
• the hump rises as c grows
• the interval ratios converge on 8.7210972
• the largest period included is a whole number between 4 and 7
• the largest period listed is a whole number between 8 and 16
• the largest period marked is a whole number between 3 and 8
• the largest period searched numerically is a whole number between 5 and 10
• the last ratio measured is Feigenbaum's constant to within a twentieth
• the main cascade's gaps shrink by Feigenbaum's constant
• the next superstable parameter is bracketed where the cascade predicts
• the number of doublings found is a whole number between 4 and 7
• the number of doublings is a whole number between 3 and 5
• the number of mergings is a whole number between 3 and 5
• the option is one this view reads
• the orbit piles up beside each image of the turning point
• the orbit stays between the first two images of the turning point
• the orbit takes several steps to pass through the channel
• the orbits shrink by 4.018 per doubling
• the period is a whole number between 3 and 8
• the period of the window the strand belongs to is a whole number between 1 and 9
• the period-three window has one superstable parameter of period three
• the points kept per column is a whole number between 10 and 400
• the quiet phases lengthen as the inverse square root of the distance
• the range runs upward
• the range runs upward and is wide enough to draw
• the ratios climb towards Feigenbaum's constant
• the return map has its maximum inside the range — a hump, not a slope
• the root found has exactly the period sought
• the steps drawn is a whole number between 100 and 2000
• the stretch drawn holds at least two quiet phases
• the strongly dissipative cascade approaches 4.669 and the area-preserving one 8.721
• the top is rounded like a parabola, opening downwards
• the two maps have the same number of windows up to this period
• the view is one of feigenbaum, critical, histogram, merging, channel, laminar, laminarlength, zoom, cascade3, windowcount, superstable, useq, windowsmap, rossbif, rossorbits, rossreturn, rossreturns, rossdelta, rosswindow, zordhero, zordmaps, zordconv, zorddelta, zordscale, zordalpha
• α falls as the order of the maximum rises
• δ rises with the order of the maximum

• golden slope: exactly n + 1 blocks of length 1 ×14
• slope √2 − 1: exactly n + 1 blocks of length 1 ×14
• every block of length 1 holds one of two neighbouring numbers of H ×12
• in direction (1, √2, √3) there are n² + n + 1 blocks of length 1 ×9
• with X erased, 2 blocks of length 1 ×8
• with Y erased, 2 blocks of length 1 ×8
• with Z erased, 2 blocks of length 1 ×8
• 4-cube (1, √2, √3, √7): Baryshnikov's count for blocks of length 1 ×6
• cube (1, √2, √3): Baryshnikov's count for blocks of length 1 ×6
• square (1, √2): Baryshnikov's count for blocks of length 1 ×6
• a trajectory in this table takes exactly 4 directions and no more ×3
• the path bounces correctly off side 1 ×3
• 2 to 40 jumps drawn
• 4 to 9 steps
• 40 to 160 cells a side
• a block length from 2 to 12
• a closed path is one that returns to within a tolerance
• a direction with a whole-number relation among its components has fewer
• a jump round a circle keeps the distance from the centre
• a named direction
• a named slope
• a path closing on itself was found within 1e-9 — the search at this depth found none otherwise
• a square or a triangle
• a table the family knows
• and arrives and leaves at the same angle to the radius
• and does so after twice the sum of the two whole numbers
• and it is the word a line of slope 1/φ spells, started at the corner
• and its angles add to a straight angle
• and its invariant is the same at every bounce
• and never returns exactly to where it started
• and no nearby triangle inscribed in the same three sides is shorter
• and one H for every horizontal one
• and that genus is a whole number
• and the direction along the wall is untouched
• and the path does not close before it is supposed to
• and the two rare blocks never do
• and they grow more common the further it is nudged
• and turns it by twice the angle whose cosine is one over that distance
• and with the obstacle the gap grows by far more than the distance does
• at least one drawn orbit closes, so the two behaviours are both on the page
• at the value it was aimed with
• between 10 and 200 bounces
• between 20 and 200 bounces are followed
• between 3 and 24 bounces are drawn
• between 3 and 40 chords are drawn
• each altitude meets its side between the corners
• each block's frequency is one of the arcs, largest to largest
• each bounce lands on a wall of the kind its letter names
• each jump lands as far beyond the vertex as it started before it
• each word begins with the one before
• every chord passes the same distance from the centre
• every orbit about a table with whole-number corners closes
• every point outside the table has a vertex it jumps through
• in both coordinates
• in direction (√2, √3, √5) only forty-one blocks of six appear in a million letters
• in four dimensions the count is n³ + 2n + 1
• no orbit started in the picture wanders far from the table
• nudge the third component by half a percent or more and all forty-three appear
• on the empty table the gap grows no faster than the distance travelled
• on the empty table the gap stays proportional to the distance travelled
• only the fastest letter ever comes twice running
• orbits further out take at least as long to close
• so no part of it is below the axis
• the angle of arrival equals the angle of departure at every wall
• the angles of the table add to what a polygon's angles must
• the arcs take at most three lengths
• the ball is moving and has a wall to reach
• the ball starts inside the table
• the bouncing path and the folded straight line agree everywhere along their length
• the drawn triangle has the angles it was asked for
• the escaping trajectory's invariant is below it
• the first hit is somewhere on the rim
• the irrational path enters nearly every cell of the grid
• the irrational path is followed for between 60 and 600 bounces
• the lengths are Fibonacci numbers
• the obstacle fits inside the table with room to pass
• the path is followed for a drawable number of bounces
• the rational slope is a ratio of small whole numbers
• the rational slope is in lowest terms
• the rational slope's path returns to its exact starting state
• the rational slope's word never has more than seven blocks of any length — it repeats every seven letters
• the reflected copies drawn fit inside the panel they are drawn in
• the search runs to between three and sixteen bounces
• the slope is drawable
• the square's word has n + 1
• the star closes exactly
• the star polygon is drawn in a single closed loop
• the star wraps between 1 and 6 times
• the stem is between a fifth and a whole radius deep
• the stem is between a twelfth and half the cap wide
• the surface the table unfolds into has a whole genus
• the sweep is a grid of a workable size
• the sweep produced something to compare
• the table is a triangle
• the table is one the family knows
• the trajectory is followed for between 40 and 600 bounces
• the trajectory stayed inside the table
• the trapped trajectory never touches a stem wall
• the trapped trajectory's invariant is above the stem's half-width and inside the cap
• the triangle is acute, which is when this path exists
• the Tribonacci word has 2n + 1
• the two invariants sit on opposite sides of the stem's half-width
• the two numbers bracket n times the share of H
• the two paths start very nearly together
• the view is one the family draws
• the word has one V for every vertical line crossed
• there are 7 blocks of length 6
• thirteen of the twenty-seven blocks of three appear
• two angles of a triangle, in degrees, leaving a third
• two to five starting distances outside the unit circle
• while the trajectory aimed under the half-width does go down the stem
• with the obstacle the same two paths end up hundreds of times further apart

• piece 0's mass is read off its binary digits ×1024
• at depth 2 the cover's total agrees with the closed form at exponent 1.037 ×36
• 0 right-hand choices are shared by C(10, 0) pieces ×19
• the variogram estimate is unbiased on fractional Brownian motion at D = 1.9 ×9
• after 0 steps there are two to the power 0 pieces ×7
• at the dimension itself the total is exactly one, at depth 2 ×6
• the midpoint construction and the series agree at 1/2²⁰ ×6
• f at q = 0 is the least value of qα + β(q) ×5
• at α = 0.5 the measured memory falls as β rises ×4
• the measured dimension at α = 0.5 is near 2 − α/2 ×4
• the counted slope for q = 0 is β(0) from the formula ×3
• the counted slope for w = 0.6 is 2 + log₂ w ×3
• the measured Hurst exponent at β = 0.2 is within reach of 1 − β/2 ×3
• α is minus the slope of β at q = 0 ×3
• after 6 steps there are as many pieces as the rule makes ×2
• f at q = -3 is the least value of qα + β(q) ×2
• a finite number of splits undercounts
• a shift by ⅔ or by 2/9 keeps half the pieces
• a smaller box never covers the set in fewer boxes
• a thousand samples already pin the dimension
• and above it the total shrinks to nought
• and clearly different memory
• and its limiting dimension is the curve's
• and none is larger than the support
• and the average over eight independent paths is closer
• and the point of contact is the information dimension
• and they make up the whole support
• at every level the sums of the Cantor set's pieces fill [0, 2]
• at q = 0 every piece counts once, which is the support's dimension
• at q = 1 the curve touches the diagonal
• at q = 1 the moment is the total mass, so β(1) = 0
• at the smallest boxes the thin branches look like curves; at larger boxes the slope approaches 1.7
• below a third the self-sum's length shrinks like (3λ)ⁿ
• below the dimension the total grows without bound as the cover is refined
• between 20 and 400 thousand steps
• between one and four sets this family draws
• between two and six depths, each between 1 and 14
• C + C stays the whole of [0, 2] at every level
• C and C + t are disjoint when |t| > 1
• C meets itself in every piece
• each count is exactly the number of pieces at that depth
• every pair's thickness product lies on the drawn axis
• for λ at least a third the self-sum is the whole interval
• half of all first contacts land on a few per cent of the perimeter
• how many box sizes are counted is a whole number between 3 and 6
• how many times the box size is divided is a whole number between 1 and 5
• how many times the mass is split is a whole number between 1 and 12
• how many times the rule is applied is a whole number between 2 and 6
• more splits bring the counts closer to the curve
• near w = 1 detail finer than the grid is still large and the count reads low
• near w = ½ the count carries a logarithm and reads high
• no set of points is larger than its own scaling exponent allows
• one to five exponents between −2 and 6, leaving out 1, whose sum is always the whole mass
• one to four factors between ½ and 0.9
• one to three increasing split counts, up to 1000
• splitting in two, level times, makes two to that power pieces
• the aggregate spreads far wider than the Eden cluster with the same particles
• the aggregate's mass grows like r^1.7, Eden's like r²
• the box count moves by more than three tenths across the units
• the box count reads low on rough paths and close on smooth ones
• the box level marked is a whole number between 2 and 7
• the Brownian path's counted slope is near 1.5
• the chosen cells are distinct positions of the grid
• the circulant embedding's eigenvalues are non-negative
• the comparison measure is uneven enough to have a curve
• the contraction beats the stretch, so the formula's first case applies
• the count's exponent is the curve's at the matching q
• the counted box dimension closes on the closed form
• the counted dimension closes on the one the rule forces
• the counted dimension lies strictly between a curve's and a region's
• the counted slope for q = -1 is β(-1) from the formula
• the curve is drawn finer than the smallest box counted
• the drawn boxes are the counted boxes
• the exponent is swept between 0.05 and 1 either side of the dimension
• the factor each level's raise shrinks by is between 0.3 and 0.95
• the first exponent is positive, which is what makes the orbit sensitive to its start
• the formula returns a value between a curve's dimension and a region's
• the Hausdorff dimension never exceeds the box dimension
• the horizontal contraction is the stronger one, which is what makes the carpet self-affine
• the left half is a line plus the whole graph shrunk by w
• the local dimension runs from near 1.8 at short lags to near 1.2 at long ones
• the map contracts area but does not collapse it
• the map's parameter is in the range that has an attractor
• the marked columns are the counted columns
• the masses add to one
• the measure lives on the interval or on the Cantor set
• the number of box sizes is a whole number between 4 and 10
• the number of columns is a whole number between 2 and 6
• the number of pieces and the ratio give the dimension the rule forces
• the number of rows is a whole number between 2 and 5
• the number of sampled points is a whole number between 50000 and 1000000
• the number of times the rule is applied is a whole number between 1 and 7
• the orbit stays on the attractor rather than escaping
• the radius of gyration gives a dimension near 1.7
• the raises at a fine level have the variance the construction prescribes
• the rough, long memory record's measured dimension is near 2 − α/2
• the rough, short memory record's measured dimension is near 2 − α/2
• the rows carry different numbers of cells, so the two dimensions differ
• the rows carry equal numbers, so the two dimensions agree
• the sample is finer than the smallest box counted
• the second is negative, which is what collapses the attractor onto a set of no area
• the seed of the Brownian path is a whole number between 1 and 99
• the seed of the records is a whole number between 1 and 99
• the self-sum's dimension is log 3/log(1/λ), capped at 1
• the set is one built by a rule, or there is no exponent to pivot about
• the set is one this family draws
• the share of the mass the left piece of each split carries is between 0.02 and 0.98
• the shares of the measure drawn for comparison is between 0.02 and 0.98
• the smooth, long memory record's measured dimension is near 2 − α/2
• the smooth, short memory record's measured dimension is near 2 − α/2
• the two exponents add to the logarithm of the area factor, which the map fixes
• the two records have the same measured dimension
• the variogram estimate ignores the units
• the view is one the family draws
• thickness is bridge over gap
• twenty thousand particles reach a radius of a few hundred lattice steps
• when the thicknesses multiply to at least 1 every interleaved translate meets
• while the Hurst exponent still spreads widely at a quarter of a million
• with equal shares every point scales alike
• α is minus the slope of β at q = -2

• the orbit 2/3 is found at k = 0.9 ×43
• inside the plateau at Ω = 0.4645 there are two orbits ×10
• the residue of 3/5 crosses a quarter between 0.9 and 1.05 ×8
• the nonlinearity is between 0 and 0.95 ×6
• a periodic orbit with rotation number 0/1 exists ×5
• the orbit of 0/1 closes ×5
• some parameter carries a period-2 orbit with rotation number 1/2 ×3
• the rotation number there is exactly 1/2 ×3
• inside the plateau at Ω = -0.1528 there are two orbits ×2
• the number of parameters sampled is a whole number between 60 and 900 ×2
• a whole partial quotient between 2 and 100,000
• above it they grow without bound
• above the critical line the interval opens
• above the critical line the lift turns back on itself
• and at least one is outside it
• and both boundary circles still stay where they are
• and increasing everywhere else
• and it rises rather than falling
• and the reason is that this map does not preserve area
• and the repelling one's is at least one
• and the second really is outside it
• and they turn in opposite directions
• and would change an exponent-one total a great deal, so that one does not
• at it they hold near a quarter
• at least two of the panels are inside the plateau
• below it they shrink by (1 + √2)²
• below it they shrink by the square of the golden ratio
• below the critical line every interval is a single number
• below the critical line the gap ratio is the square of the ratio
• below the critical line they shrink by the golden ratio
• below the critical strength the residues shrink to nothing
• both boundary circles stay where they are
• both maps give the same ratio
• both solve from a standing start
• doubling the range barely changes this total, so it converges
• each crossing is a fixed point of the map
• each orbit is followed for between 60 and 600 steps
• every conjugacy drawn solves and is increasing
• every observed average advance lies in the interval
• every rescaled point lies inside the drawing
• how many images are drawn either way is a whole number between 6 and 20
• in the sorted coordinates the map is the rotation, to the orbit's own resolution
• inside a plateau the rotation number does not depend on where the orbit started
• magnified twice more, the staircase looks the same
• most sampled parameters carry an orbit
• Newton's method closes at every K
• Newton's method closes at every K drawn
• Newton's method closes for every series drawn
• no orbit leaves the annulus
• no point of the annulus comes near to staying where it is
• on the critical line the gaps shrink by 2.8336
• on the critical line the silver gaps shrink by about 6.799
• once the ones take over, the golden number returns
• one fixed point is a centre and the other a saddle
• one orbit attracts and the other repels
• one to four nonlinearities below one
• one to three strengths
• outside, the rotation number has moved off the rational
• some start near the middle advances at the golden rate
• the advance sticks at rational values over whole stretches of starting momentum
• the angle is a number or one of the named constants
• the angular shift changes sign across the annulus at every angle
• the attracting orbit is most strongly attracting in the middle of the plateau
• the attracting orbit's multiplier is at most one
• the boundaries still turn in opposite directions
• the circle wrinkles more as k grows
• the closest returns shrink by 1.2886 on both maps
• the conjugacy's strip is never wider than the map's
• the conserved quantity stays put along every orbit drawn
• the critical maps are flat at θ = 0
• the crossings close in on 0.9716
• the drawn images account for less than the whole circle
• the drive is between 0 and 1
• the envelopes sandwich the lift
• the exponent the lengths fall at is between 1.05 and 4
• the first parameter really is inside the plateau
• the gap ratios or the return ratios
• the gaps take at most three distinct values, as the three-distance theorem says
• the golden divisors never fall below about one over k
• the kick is between 0 and 1.5
• the largest nonlinearity is between 0.4 and 1.2
• the largest period is a whole number between 2 and 9
• the length of the orbit is a whole number between 200 and 3000
• the lengths sum to something finite
• the locked share of the parameter axis grows with the nonlinearity
• the loop and its image cross exactly twice
• the loop and its image enclose the same area
• the lower end never exceeds the upper
• the lower parameter is between 0 and 1
• the map locks onto one step in two over an interval of parameters
• the map preserves area everywhere on the annulus
• the measured curve is flat over a stretch at more than one rational
• the number of drives sampled is a whole number between 40 and 300
• the number of nonlinearity levels is a whole number between 6 and 40
• the number of parameters per level is a whole number between 40 and 320
• the number of starting points is a whole number between 100 and 4000
• the number of steps drawn is a whole number between 10 and 60
• the number of steps is a whole number between 3 and 120
• the option is one this view reads
• the orbit leaves no large gap
• the orbit repeats exactly when the angle is rational
• the outward push is a sensible size
• the parameter reaches the intended rotation number
• the period is a whole number between 1 and 4
• the perturbation is a sensible size
• the perturbation is between a fiftieth and two fifths
• the plateau has positive width
• the ratio of consecutive lengths rises towards one, which is the smoothness condition
• the returns shrink by about 1.29 per Fibonacci step
• the rotation number never falls as the parameter rises
• the rotation number p/q is between nought and one
• the rotation per period, plus one is a whole number between 1 and 5
• the search interval contains the parameter
• the tangent map round the orbit preserves area
• the two families approach the same silver number
• the two orbits are closer together near the edge of the plateau
• the upper parameter is between 0 and 1
• the view is one the family draws
• the window runs upward and is wide enough to draw
• two to six leading twos
• up to four settings, each below the critical line
• well above it most intervals are open
• with no nonlinearity the curve is a straight line and has no plateaus to speak of

• 1/2 is purely periodic, with no digits before the repetition begins ×190
• T2: bin 1 matches the arcsine share ×90
• bin 1 holds the share the density says it should ×60
• at r = 4 the 0-th iterate has 2^0 folds ×21
• every fraction in lowest terms with denominator 2 repeats with the Fibonacci period of 2 ×19
• T2 composed 1 times has 2 fixed points ×16
• T2 composed 1 times is T2 ×16
• step 0 is cos(3^0 θ) ×13
• returns after 1 steps against the trace ×10
• the orbits of exact period 1, walked and by inversion ×10
• the points of exact period 1 make whole orbits ×10
• walks of length 1 from the start state are the admissible strings ×10
• sampling finds every admissible word of length 1 (the golden ratio φ) ×9
• sampling finds every admissible word of length 1 (β = 1.8) ×9
• sampling finds every admissible word of length 1 (β = 5/2) ×9
• at level 1 the admissible intervals number F(3) = 2 ×8
• the number of steps is a whole number between 2 and 400 ×7
• every point over 1 comes back after 1 doublings ×6
• the trace is 1 + (1 + √2)ⁿ + (1 − √2)ⁿ at n = 1 ×6
• 0.00000 returns to itself after 2 steps ×5
• a full two-branch fold has 2^1 points fixed by its 1th power ×5
• after 0 doublings the overlap is within 1/2^2 of a sixth ×5
• both maps have the same number of points of period dividing 1 ×5
• 0.6154 is fixed at r = 2.6 ×4
• after 0 doublings the lowest frequency present is 1 ×4
• T2(cos θ) = cos 2θ ×4
• the random point visits quarter 1 a quarter of the time ×4
• the total energy of f composed 0 times is a quarter ×4
• in the window of three, at 3.8300, the estimate is log of the golden ratio ×3
• the folds of f^6 at r = 3.5, sampled and from the preimages of 1/2 ×3
• the longest word counted is a whole number between 3 and 10 ×3
• the numerator of the starting point is a whole number between 1 and 10000000 ×3
• the parameter is between 0 and 2 ×3
• at r = 2.4 the error shrinks by the slope at the fixed point each step ×2
• the longest iterate counted is a whole number between 6 and 22 ×2
• the number of bins is a whole number between 8 and 80 ×2
• the number of steps drawn is a whole number between 3 and 20 ×2
• the number of times the map is applied is a whole number between 2 and 5 ×2
• the orbit at r = 2.4 converges far enough to measure a rate ×2
• √2 begins 1.0110101000001001111… in binary
• √2 is fixed by the Newton step
• 0, 1/3 and 1 are a cycle of three
• 1/3 repeats every eight digits
• 1/3's share of 1s settles at a quarter, not Parry's value
• 100 to 800 values of r
• 2 has order 2·3^(k−1) modulo 3^k
• 2/7 repeats every sixteen
• a base above one up to the top end
• a base below the lower end
• A covers A and B, B covers all three, C covers B and C
• a cubic with the requested critical values exists
• a fraction strictly between 0 and 1
• a hump between 0.45 and 1 and a dip between 0 and 0.55
• a new crossing has a period above one that divides 2
• a point with no two 1s together never enters the top quarter
• after 3^k the base-6 digits are zero for about a sixth of 3^k
• after the computed orbit is dead the exact one is still moving
• and carries the right end to the right end
• and it never reaches zero, because an odd denominator cannot be halved away
• and never grows
• and not before
• and shrinks every distance by a factor under one
• and stays there, because zero is fixed
• and the constant the ratio approaches is 1/(2√2)
• and the repeat found really is a repeat
• and then separate visibly, at a step the figure marks
• at 1,000 digits more than 99% of strings are within 0.05 of a half
• at r = 4 the entropy is ln 2 and in the window of three it is ln φ
• at r = 4 the folds double and the estimate is log 2
• at the end of the doublings the signs θₖ are the Thue–Morse signs (−1) to the number of 1s in k
• below the end of the doublings the deeper estimate is the smaller
• between 6 and 60 steps are taken
• between one and six logistic parameters from 3 to 4
• between one and three logistic parameters, each with an attracting fixed point
• block counts of every length are within the spread of random digits
• deleting the first digit of an admissible word leaves it admissible
• double precision follows the orbit for about fifty steps
• double precision tracks the digits for about seventy-five steps
• each block's frequency sits within sampling error of Parry's
• each density has total mass one
• each drawn level curve is a single connected piece
• each letter of the itinerary is the next binary place of the starting point
• each Newton error is the square of the last over twice the current guess
• each point of the cycle is carried to another
• each point of the cycle lies in its letter's piece
• each step of a closed word's point lands in that word's piece
• every base-6 digit between one term's end and the next term's start is 0
• every bin is visited
• every corner of the staircase lies on the curve or on the diagonal
• every crossing of the map is also a crossing of the composed map
• every departure lies inside the drawn range
• every exact iterate is a whole numerator over the same denominator
• every orbit's share settles on the predicted frequency
• every piece is stretched
• every start reaches the same point
• every word the map writes passes Parry's test
• five Newton steps from 1.9 reach twelve decimal places
• for the tent map of slope s the kneading series first vanishes at 1/s
• from the thirtieth digit on the running share stays inside the drawn range
• how many steps of the orbit are drawn is a whole number between 3 and 12
• in base the tribonacci constant, 1 = .111
• in base φ no two 1s are ever consecutive
• in base φ the allowed words are counted by the Fibonacci numbers
• in base φ the digit 1 appears with frequency 1/(1 + φ²)
• in base φ, 1 = .11
• its denominator is a whole number between 3 and 10000000
• just after each gap the share of zeros is well above a sixth
• no point is named by two closed words
• no two 1s in a row, as every base-φ expansion must
• on an odd ring exactly the even-weight words lie on cycles
• one point drawn per step, and the start
• r is a decimal string between 3 and 4
• random digits sometimes put every statistic on one side
• T₂∘T₃, T₃∘T₂ and T₆ agree everywhere
• the 2 new crossings of period 2 are whole orbits of 2 points
• the average of one long orbit is the average over the whole interval
• the base has a finite expansion of 1, so its graph is finite
• the base is one the family knows
• the binary blocks of α are as even as fair coin tosses would make them
• the change of coordinate fixes the left end
• the change of coordinate is increasing, so it is reversible
• the column count is a whole number between 20 and 241
• the computed orbit reaches exactly zero inside the drawn window, having run out of binary places
• the contraction is one the family knows
• the coordinate change carries the tent map to the logistic map
• the coordinate is one whose invariant density this figure knows
• the counts grow by a factor of β per letter
• the cycle closes after the word's length
• the cycle is one the family draws
• the deeper estimate never falls as r rises
• the degree is a whole number between 2 and 5
• the denominator is odd, or the exact orbit reaches zero as surely as the float one does
• the digits agree at two precisions, so rounding has not reached them
• the digits of 1 add back up to 1
• the digits written reconstruct the start to within β to the minus the number of steps
• the drawn region is covered by computed cells
• the entropy from the kneading series agrees with the growth of the laps
• the entropy never falls as r rises, to within the truncation of the series
• the error after n steps is inside k to the n times the error at the start
• the exact orbit closes up inside the drawn window
• the exact shares add to one
• the fixed point attracts, so there is a rate to measure
• the folds grow by the golden ratio at the last step
• the frequency of nonzero digits is the density's mass on the right piece
• the graph's growth rate is the base
• the highest frequency drawn is a whole number between 16 and 96
• the histogram is consistent with the arcsine law at the 0.1% level
• the image intervals stay few enough to follow
• the k-th term's base-6 digits begin near (1 − log₆3)·3^k
• the kneading series at the end of the doublings stays positive as far as 256 terms can follow it
• the left piece covers the right, and the right covers both
• the longest cycle counted is a whole number between 2 and 8
• the longest word counted is between two and eight
• the longest word is a whole number between 6 and 14
• the lower parameter is between 3 and 4
• the map is one the family knows
• the map is one this figure knows
• the map itself brings the point back
• the map itself has a fixed point
• the map sends the interval into itself
• the nested picture takes four numbers
• the number of binary places the starting point has is a whole number between 3 and 8
• the number of drawn steps is a whole number between 2 and 400
• the number of levels is a whole number between 4 and 10
• the number of stages is a whole number between 3 and 7
• the number of times the map is composed is a whole number between 2 and 4
• the option is one this view reads
• the orbit approaches the fixed point exactly when the slope is shallow
• the orbit at r = 3.83 settles on a cycle of three
• the orbit of 1/3 lives in the two middle quarters
• the orbit of 1/3 never enters [0, ¼)
• the orbit starts inside the interval, or at 1
• the orbit stays inside the unit interval
• the orbit's histogram matches Parry's density away from the steps
• the point a repeating word names has that word as its itinerary
• the point found by bisection is fixed
• the points returning after 4 steps are as many as the trace of the matrix power
• the quadratic curve is on or off
• the random point's running average reaches a quarter
• the range runs upward
• the remainder stays in [0, 1)
• the repetitions are drawn no closer than the screen can separate
• the run ends
• the running share stays within the spread of a random sequence
• the share near a half grows with the length
• the size falls by the factor q − 1 at each step
• the slope at the fixed point is 2 − r
• the solved point comes back after exactly the word's steps
• the starting point has between three and eight binary places
• the starting point is between 0 and 1
• the starting point is inside the interval
• the steps drawn is a whole number between 3 and 14
• the straight pieces of the composed map are the allowed words of 4 letters
• the T3 histogram is consistent with the arcsine law
• the transported orbit is the orbit of the transported point
• the transported orbit obeys the tent map's own recurrence
• the tripled digits give cos 243
• the two agree exactly for at least eight steps
• the upper parameter is between 3 and 4
• the view is one the family draws
• the word is not a shorter word repeated
• the word is two to eight letters of L and R
• the word, read round, never puts L after L
• the words of each length and the points of that period are equally many
• tripling stops well before the digits run out
• two depths, the deeper at most 20
• two logistic parameters above one
• two points are at most k times as far apart after the map as before
• under doubling the orbit spreads evenly; under multiplication by six it piles up near nought
• x² and x² − 1 composed in the two orders differ
• α begins 0.0000101010111000111… in binary

• step 1 follows the rule ×178
• every one of the 2 patterns of length 1 occurs exactly once ×12
• the orbit of 27 reaches one ×7
• −1 is a fixed point of T
• a positive starting number
• a power of two is never a power of three
• a starting number is a whole number between 3 and 1000000
• across a large sample of starts, half the steps are odd ones
• an integer agreeing with −1 to k digits takes k odd steps first
• and 27 is the famous long one
• and by the largest run drawn it is above nine tenths
• and it is the one at one
• and it then comes down to 1 all the same
• and no two residues give the same parity string
• and reaches a few thousandths within the range drawn
• and the average fall per step is the value the pairing predicts
• at least one pattern gives a whole number
• between two and four odd offsets, each at most eleven
• each convergent beats every fraction with a smaller denominator
• each new digit of x is decided by the next parity
• each row's parity string is the row above shifted left by one
• every number in the program factors over the first ten primes
• every number the reverse tree reaches does come back to one
• every number up to 2,000 is hit
• every orbit trends downwards in the logarithm
• every pattern of the length was solved
• every peak is held exactly in double precision
• every positive whole number has exactly one preimage
• in parity coordinates T is the doubling map
• it halts at 5 to the power a·b and nothing else
• its size after k steps is 3^(odd steps) / 2^k times the start, to within rounding
• most parity strings shrink the number by the end of the run
• no pattern with a negative denominator gives a positive whole solution
• no two numbers go to the same place
• one is already at one
• one length per start
• past a hundred thousand every record's exponent lies between 1.7 and 2.1
• path records arrive at a steady rate per decade
• Q sends the numbers below 2^K to all 2^K parity strings, each once
• several cycles are found
• six to thirteen binary digits
• small factors
• so the residues and the parity strings match one for one
• the average of the γ-th power of the step is one at γ = 0 and γ = 1
• the chance of climbing to x times the start falls like 1/x
• the continued fraction of log₂3 starts 1, 1, 1, 2
• the denominator is always odd, so the solution stays where the map is defined
• the depth of the tree is a whole number between 3 and 12
• the exponents of the powers of two are the primes, in order
• the first path records are 3, 7, 15, 27, 255, 447, 639, 703, …
• the integer follows its pattern for all k steps
• the largest number of steps is a whole number between 4 and 15
• the largest start is a whole number between 20 and 4000
• the longest pattern length checked is a whole number between 6 and 14
• the longest pattern searched is a whole number between 8 and 20
• the map is a bijection at every length checked
• the multiplication program halts
• the multiplier condition predicts the drop, on a large member of the class
• the number of convergents listed is a whole number between 5 and 10
• the number of steps is a whole number between 3 and 7
• the orbit of 8 has not closed up within the steps drawn
• the orbit's peak is the record
• the ordinary rule has exactly one cycle at these lengths
• the ordinary rule is among those searched
• the parity of the first steps depends only on the start modulo two to the k
• the parity vector of T(x) is the parity vector of x shifted by one
• the pattern length is a whole number between 4 and 10
• the relative gap shrinks along the list
• the rule with minus one has at least three, which no drift argument distinguishes
• the run reaches several powers of two
• the share provably driven down grows with the number of steps allowed
• the starting number is a whole number between 1 and 100000
• the tree is still growing at the depth drawn
• the view is one the family draws

• b0 agrees with the value computed by hand ×6
• a longer wait only ever decides more
• along 1/q the ratio has settled
• and after sixteen thousand terms it is still far above the counting estimate
• and always straddles 1.5066
• and neither is four tenths
• and so is minus one
• at the root of the p/q bulb the fixed point's multiplier is exactly e^(2πip/q)
• both rays of each pair end, at depth 160, within two hundredths of the bulb's root
• each pair's rays end together, beside a centre of exact period n
• every angle finds a partner
• every bulb up to period forty is within a fifth of sin(πp/q)/q²
• every bulb up to period thirteen turns its cycle by its own fraction
• every bulb up to period thirty is drawn
• every term lowers the bound
• every vanishing coefficient has an index divisible by 4
• exactly one cycle of the doubling map turns p places of q
• exactly two of its angles are neighbours, one step of 1/(2^q − 1) apart
• for every fraction with denominator up to fourteen the landing angles are P01 and P10
• for every pair of neighbouring fractions, the largest bulb between them is their mediant's
• most pairs end a thousand times closer to each other than to any other ray
• of the 132 non-crossing pairings of the twelve angles, two also clear the chords of periods two and three
• on period four, 6/15 is refused 7/15 and 8/15 and joined to 9/15
• one is not
• round the fixed point, each step of the cycle advances p places of q
• summed over every period the limbs take all the angles
• the 235 chords to period eight cross nowhere
• the bulb is p/q in lowest terms with q from 3 to 9
• the bulb's outline is found at every multiplier
• the centre of the p/q bulb has period q, for every bulb up to period twelve
• the coefficients shrink a little faster than one over m
• the critical point is in this Julia set exactly when its parameter is in the Mandelbrot set
• the escape time separates into bands
• the first coefficients to vanish are b4, b8, b12, b16, b24, b32
• the fraction is in lowest terms with q from 2 to 7
• the iteration cap is a whole number between 20 and 400
• the Julia parameter is a complex number
• the landing angles differ by exactly 1/(2^q − 1)
• the limbs' angles add up, period by period, to φ(q)/(2^q − 1)
• the option is one this view reads
• the pairs of every period number its centres, and the pairs on one cycle number the satellite bulbs
• the pairs that share a cycle end far wider apart than the pairs that do not
• the ray of angle 1/6 approaches i geometrically
• the ray of angle 1/7 approaches the root like one over the depth
• the refusals grow with every period, past twenty for each pair at period twelve
• the sample count is a whole number between 60 and 700
• the set is the Mandelbrot set or a Julia set
• the span is between 0.0005 and 4
• the undecided cells straddle the area
• the undecided share shrinks with every refinement
• to period seven, every pair's rays end closer to each other than to any other ray, beside its own centre
• towards the golden angle the bulbs outgrow the law; along 1/q they do not
• zero is in the Mandelbrot set

• doubling 1 times returns 0/1 to itself ×120
• at r = 4, ε = 0.00 the measured exponent is λ + ln|1 − 2ε| ×74
• at β = 0.30 the root-mean-square grows faster than a typical run ×17
• of total length (2/3)^0 ×11
• after 0 steps there are 2 to the 0 surviving pieces ×6
• slope 3: box-counting dimension equals 1 − κ/λ ×4
• the number of steps is a whole number between 10 and 200 ×3
• a one-way pull of 0.35 fails and 0.6 succeeds
• a periodic point lies within the interval
• a perturbed q-twisted state returns to itself exactly when cos(2πq/N) > 0
• a random start settles into a twisted state with neighbouring phases differing by the same amount all round
• a random start survives s/(s − 2) steps on average
• a rational turn has only its periodic points
• a run of a million steps gives Viswanath's constant
• a tenth of plus signs already makes it grow
• a twisted state's average phase arrow has length nought
• above the locking value the chain is one cluster
• after 57 steps every picture is turned through half a turn about the corner
• an irrational turn is transitive and nothing else
• and at the step z nears q, z and p are at least half the far distance apart
• and below the onset of chaos it is negative
• and the centre settles on Viswanath's constant
• and the centre stays put within a spread
• apart, bursting, and locked
• as the coupling falls the clusters only multiply
• at full chaos λ is ln 2
• at r = 4 the exponent is log 2, which is exactly computable
• at twenty steps the sampled mean square matches the exact one
• at ε = 0.776 the ring of five locks and the ring of six does not
• below the threshold the residual order falls like one over the square root of N
• below the threshold the runs shrink, above it they grow, at it they do neither
• doubling on the line is sensitive and nothing else
• each run of a thousand steps grows at close to Viswanath's rate
• every chain locks 3 per cent above max |Sₖ| and fails to 3 per cent below
• every pixel's cycle length divides the period
• every run settles into a twisted state
• Gaussian frequencies: incoherent at weak coupling, ordered at strong
• just above the locking coupling every oscillator keeps one frequency; just below, the chain splits
• more plus signs never slow it down
• mutual coupling locks from a quarter
• on a 61 × 61 grid the cat map returns every picture after 30 steps
• on a 74 × 74 grid the period is 114
• one-way coupling locks from a half
• order appears near Kuramoto's threshold 2/(π g(0))
• quadrupling the steps doubles the spread
• repeating the plus step alone grows fastest
• rings of up to five can lock and rings of six or more cannot
• the cat map's period is half the Fibonacci numbers' Pisano period
• the column count is a whole number between 40 and 1200
• the cycles account for every pixel
• the dense orbit enters the interval
• the dense orbit later comes near the far point
• the direction piles up sharply rather than spreading evenly
• the doubling map has all three
• the empirical law of Kₗ/√N at a thousand oscillators is within 0.05 of Kolmogorov's everywhere
• the exact mean square has grown
• the first coupling is below the threshold of a quarter and the second above it
• the largest period is a whole number between 2 and 8
• the locking coupling over the square root of N approaches √(π/2) ln 2
• the lower parameter is between 0 and 4
• the mean square grows like φ, so its root grows like √φ
• the measured escape rate is log(s/2)
• the median run has shrunk
• the number of strips is a whole number between 4 and 64
• the orbit enters every one of the 16 strips within 400 steps
• the orbits separate exactly when the parameter is past the onset of chaos
• the parameter is between 0 and 4
• the period equals 3N exactly at N = 10, 50 and 250 in this range
• the period never exceeds three times the grid size
• the picture comes back at step 30 and not at the steps drawn before it
• the range runs upward and is wide enough to draw
• the separation grows by φ² = 2.618 at every step until it saturates
• the share of locked oscillators is the share with |ω| below K r
• the slope is above 2, so the middle escapes
• the spread of the estimate narrows as the runs lengthen
• the starting point is between 0 and 1
• the three couplings sit below the average threshold, just above it, and above the fixed point's
• the truth lies strictly between the naive guesses
• the two orbits begin the stated distance apart
• the two starts are close together
• the typical rate crosses one at the threshold
• the upper parameter is between 0 and 4
• the view is one of lyapunov, dense, periodic, banks, table, cantor, survival, transient, kg, randfib, randfibrate, randfibnaive, randfibangle, randfibbias, betaruns, betarate, betamean, betaspread, betapatterns, syncpair, syncsweep, syncdrive, syncbubble, syncring, kurhero, kurcurve, kurfinite, kurtime, kurlocked, kurgauss, kchero, kcsums, kctest, kcsqrt, kckolm, kcclusters, krhero, krbasins, krstable, krreach, krslips
• the weak pair stays apart and the strong pair locks
• the winding number stops changing early in the run
• the winding number's variance grows in proportion to the ring's length
• weak coupling leaves the phases scattered and strong coupling gathers them
• well above the threshold the measured order matches √(1 − 2γ/K)
• with enough neighbours every start synchronises; with two, under three in five do
• with minus signs only the sequence cycles and does not grow
• with plus signs only it grows like the golden ratio
• z and p start inside the same small interval

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