Small cases lie — page 6
Algebra
How a polynomial breaks modulo the primes
Reduce x³ − 2 modulo a prime and it factors: into three linear pieces for some primes, one linear and one quadratic for others, not at all for the rest. Over the primes up to twenty thousand those three patterns occur a sixth, a half and a third of the time — exactly the shares of the identity, the flips and the rotations in the symmetry group of a triangle, the group that permutes the three cube roots of 2. A polynomial's factorisations modulo primes are a census of its Galois group.
Analysis
A geometric series whose ratio is a matrix
1 + r + r² + … adds to 1/(1 − r) when r is smaller than one. Put a matrix in place of r and the same formula holds, with the inverse matrix in place of the fraction — but what must be smaller than one is not the matrix's size. It is its largest eigenvalue. A matrix whose eigenvalues are 0.9 and 0.8 can stretch vectors ten times over before its powers begin to shrink, and the series still converges, after a detour the eigenvalues say nothing about.
Algebra
Thirty-one moves from solved
Parity settles which half of a sliding puzzle's arrangements can be reached and is silent about how far away any of them is. Searching every reachable arrangement of the three-by-three tray answers the second question exactly — two arrangements sit thirty-one moves out — and parity turns up again, this time as a law about distance.
Discrete
The order decides the colours
The simplest way to colour a graph is to take the vertices one at a time and give each the first colour its neighbours are not already using. It never needs more than one colour beyond the largest degree — and on a graph that needs only two colours it can be made to use as many as there are vertices on a side, depending on nothing but the order it is handed.
Algebra
The biggest box built from signs
Fill a square table with plus and minus ones and ask how large its determinant can be. The columns all have the same length, so the answer is a box with fixed edges — largest when every corner is square, which is possible only when the size is a multiple of four.
Analysis
An ellipse, not a disc
A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.
Applied
The cube that takes every corner
The simplex method is fast on every program anybody meets in practice. In 1972 Victor Klee and George Minty squashed a cube so that the method, choosing the steepest edge each time, visits all of its corners — 2ⁿ − 1 moves in n variables, with the optimum one edge from the start.
Discrete
Every crowd holds a bowl or a dome
Among enough points in the plane, some k of them always bend upward like a bowl or some l bend downward like a dome. The number that forces it is a binomial coefficient, it is exactly right, and it proves that every large enough crowd contains a convex polygon — with a bound nobody could lower to the true answer for eighty years.
Discrete
The longest climb of a shuffle
Erdős and Szekeres guarantee that any n distinct numbers hold a climb or a fall of √n, and there are orders that allow nothing more. Shuffle a deck at random instead and the longest climb is almost exactly 2√n — with fluctuations of size n to the power one-sixth, distributed exactly as the largest eigenvalue of a large random matrix.
Number
How evenly the fractions spread
List every fraction between nought and one with denominator at most n, in order. They spread across the interval almost evenly, and how fast the unevenness shrinks as n grows is — exactly, provably — the Riemann hypothesis. The link runs through a second fact: set the fractions round a circle and add them as arrows, and what is left is a whole number.
Algebra
Two graphs the eigenvalues cannot tell apart
A graph's matrix has eigenvalues, and they count a surprising amount of the drawing: its edges, its triangles, every closed walk of every length. They do not count everything. A star with four arms and a square beside a lone point have the same eigenvalues exactly, although one of them is in two pieces — and on six points ten of the 156 graphs have a twin of this kind.
Analysis
A slope for a curve that is no function
The folium x³ + y³ = 3xy loops back over itself, so no formula y = f(x) describes it, and yet at almost every point it has a perfectly good tangent. Differentiating the equation as it stands gives the slope, −(∂F/∂x) ÷ (∂F/∂y), and the only points where that fails are the ones where the curve turns vertical or crosses itself — which are exactly the points where it stops being a graph.
![The slope of x² sin(1/x): discontinuous at nought, and never jumping. The derivative of x^2 sin(1/x) on [−0.12, 0.12], with the value at nought marked and the level 0.5 crossed 77 times.](/f/the-slope-of-x2-sin-1-x-discontinuous-at-nought-and-never-60857308.svg)
Analysis
A slope can swing but never jump
A function can have a slope at every point without that slope changing continuously: x² sin(1/x) has slope nought at the origin and a slope that swings between −1 and 1 however close to the origin it is taken. What a slope cannot do is jump. Darboux proved in 1875 that a derivative takes every value between any two of its values, so a step is never a derivative — and the only way a slope can be discontinuous is by oscillating.
Applied
Each user pays for its own last link
Several users must be connected to a source, and the cheapest network that does it is a tree. Dividing its cost so that no group of users would rather build its own looks like a hard search, and it has a one-line answer: each user pays for the link that joins it to the tree on its way to the source. No group is ever overcharged — while the average over orders of arrival, the rule that settles so much else, can charge a pair more than its own connection costs.
Computation
Every power of x that draws a hyperoval
In a plane of order 2^h, the graph of x^k plus two points at infinity is sometimes a hyperoval — as many points as a plane allows with no three in line. Searching every exponent in every plane from order 4 to 4096 finds hundreds that work, and once six symmetries of the problem are applied they fall into exactly the families already known: the conic, the translation curves, Segre's x⁶ and Glynn's two. Whether that list is complete in every order is open.
Geometry
The ball that is largest in five dimensions
A disc of radius one has area π, a ball of radius one volume 4π/3, and in each further dimension the unit ball grows — until five, where its volume is 5.264, after which it shrinks towards nothing. The recursion that shows it is the ring dissection of the disc, done one dimension at a time, and the shrinking is not the ball getting small: it is almost all of a high-dimensional cube lying outside the ball, and almost all of the ball lying in a thin rind at its surface.
Number
The row that proves a prime
Fermat's little theorem can be fooled: 561 passes it for every base and is not prime. Thread necklaces with a fixed number of black beads instead of any colours at all, and the count becomes a statement about a whole row of Pascal's triangle — every middle entry of row n is a multiple of n exactly when n is prime — which no composite can fake. Written as polynomials it is (x + a)ⁿ = xⁿ + a, and cut down to size it is the first proof that primes can be recognised in polynomial time.
Number
How often two generates every remainder
For some primes p, the powers of 2 run through every non-zero remainder before repeating; for others they get stuck in a smaller cycle. Among the primes up to two million the first kind make up 37.43% — and Artin predicted 37.40% in 1927 by treating each way of getting stuck as an independent accident. For the base 5 the same prediction is wrong, because two of the accidents are secretly the same one, and nobody can yet prove that 2 works for infinitely many primes at all.
Analysis
No integrand sits on the border
The area under 1/x out to infinity is infinite and the area under 1/x² is finite, so it is natural to look for the dividing line. There is none. Divide any divergent integrand by its own running integral and it still diverges, more slowly; divide any convergent one by the square root of its tail and it still converges, more slowly. Paul du Bois-Reymond proved in 1873 that no single integrand can separate the two families — and the slow ones are slow beyond imagining.
Computation
A third reached only in the limit
No compass-and-straightedge construction trisects every angle. But a quarter, plus a quarter of a quarter, plus a quarter of that, and so on, adds up to a third — and each of those pieces is two bisections away. So an angle can be trisected by bisecting forever, every stage exact and the shortfall shrinking to a quarter each time. The construction never ends, and that is precisely what Wantzel's proof forbids: a construction is a finite thing, and a third of a general angle is only reached in the limit.
Discrete
The fewest triangles an edge density allows
Half of all possible edges can be drawn without a single triangle. One more, and triangles appear — not one but several at once. Push the density further and the question becomes a curve: for every share of edges, the fewest triangles a large graph can hold. The answer is a string of scallops, touching a simple parabola at the densities of the balanced multipartite graphs and bulging above it between them. It was guessed in the 1980s and proved in 2008 by a method that turns counting into positive-definite matrices.
Geometry
Six numbers that need five pentagons
In 1638 Fermat wrote that every whole number is a sum of three triangular numbers, four squares, five pentagonal numbers, six hexagonal numbers, and so on for every polygon — and that he had a proof he would not write down. The claim is true; Cauchy proved it in 1813. What the claim hides is how unequal the cases are. Triangles and squares need their full count infinitely often. For pentagons, only six numbers ever need all five — 9, 21, 31, 43, 55 and 89 — and from hexagons on, two apiece.
Geometry
Fifteen numbers decide every number
Lagrange proved that x² + y² + z² + w² takes every whole value. Ramanujan asked which other sums of four weighted squares do, and in 1917 listed fifty-five. One of them is wrong: x² + 2y² + 5z² + 5w² misses 15. The mistake points at a theorem found eighty years later — to know whether such a form reaches every number, it is enough to check that it reaches 1, 2, 3, 5, 6, 7, 10, 14 and 15. Nine checks decide infinitely many cases, and the nine numbers come out of an escalation any one can run.
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.