Generator

φ as a continued fraction

A generator in the number library, called 47 times across 11 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

continued is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

φ as a continued fraction. The nested fraction, one quotient per step, descending to the right.

Lagrange's method: a cubic for each complete quotient of ∛2

The first 600 terms of the continued fraction of ∛2

Continued fractions of √2, e, π and ∛2 side by side

How close each of the first 600 convergents of ∛2 comes, for its size

How often each term appears in ∛2, against the Gauss–Kuzmin law

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

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

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Number

A cube root that looks like chance

The continued fraction of √61 repeats after eleven terms, and every square root's must. The cube root of 2 obeys no such law. Computed exactly, by a method of Lagrange's that never leaves the whole numbers, its terms run 1, 3, 1, 5, 1, 1, 4 … and throw up a 534 at the thirty-fifth place and a 7,451 at the 571st — and in every statistic anyone has measured, they look like the terms of a number picked at random.

Number

A fraction that never closes

Euclid's algorithm throws away everything except the number of squares it peeled at each step. Those counts are a second name for the number it started from — one that terminates exactly when the ratio is a ratio.

Number

A method that is allowed to miss

Bhāskara's cyclic method solves x² − Dy² = 1 by aiming at the wrong target. It keeps a pair a, b with a² − Db² = k for some small k, combines it with a helper chosen so that k can be divided out, and repeats until k is 1. For D = 61 it reaches the ten-digit fundamental solution in 13 steps, where walking the convergents of √61 takes 22 — and for every D up to 100 it is faster.

Number

Every fraction, exactly once

Take two fractions, add the tops and add the bottoms. That is not how fractions are added, it is not an average, and repeating it produces every positive rational exactly once, already in lowest terms.

Number

How close a fraction can get

Drop eight points into seven boxes and two of them share. That one line, applied to the multiples of an irrational number, proves that every irrational has infinitely many astonishingly good rational approximations — and no construction is needed anywhere.

Number

One solution that makes all the others

The equation x² − 2y² = 1 has infinitely many whole-number solutions, and every one of them is a power of the smallest. The multiplication that produces them is what multiplying two numbers of the form a + b√2 comes to when the √2 terms are collected — so an equation about a hyperbola turns out to carry a group.

Number

Sixty needs two digits and sixty-one needs ten

The smallest solution of x² − 60y² = 1 is x = 31. For x² − 61y² = 1 it is x = 1,766,319,049. The jump is not an accident of 61: the solution is exactly the product of one period's complete quotients, so its size is set by how long the continued fraction takes to come home — and for the cattle problem Archimedes is said to have posed, that product has 103,273 digits.

Number

The square that cannot shrink

The usual proof that the square root of two is irrational is about even and odd numbers. There is a proof about squares instead, in which a supposed solution is folded into a smaller one — and the folding is a drawing.

Number

Two families of solutions, and a box that holds both

Replace the 1 in Pell's equation by 7 and x² − 2y² = 7 still has infinitely many solutions — but they fall into exactly two families, each one an orbit of the same multiplication, and every family has a member inside a box whose size is fixed in advance. How many families there are is then a count of factors, and 3 has none.

Number

When minus one can be reached

x² − Dy² = 1 always has a solution. Put −1 on the right and it sometimes does and sometimes does not. A remainder rules out most D at a glance, every prime of the form 4k + 1 is guaranteed one, and between the two lies a set of D decided by arithmetic several layers deep — of which, a million numbers in, 78% are solvable, on the way to a limit of 58% that was proved only in 2022.

Number

Why the expansion has to repeat

The continued fraction of √61 runs 7; 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14 and then starts again. It must: each step's state is a pair of whole numbers trapped in a small band, and only 14 pairs fit. The expansion of √61 visits 11 of them in a cycle, the other 3 form a cycle of their own, and the period reads the same backwards before its last term, which is twice the first.

The whole library · What the figures prove