Fermats little theorem
Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.
Numbers that wrap
A clock does arithmetic. It has finitely many numbers, addition never leaves it, and multiplication behaves entirely differently depending on one property of the size of the dial.
Necklaces that prove a theorem
Thread five beads in two colours, thirty-two ways. Two of them are all one colour; the other thirty fall into rings of five. That count, and nothing else, is Fermat's little theorem.
Every element is a power of one of them
Pick the right element of a finite field and its powers run through every other non-zero element exactly once before returning to one. Multiplication becomes addition of exponents, and a table of q − 1 entries replaces the whole multiplication table.
Three in a row on the number line
Colour the numbers one to eight in two colours and it can be arranged that no three equally spaced numbers agree. Add the ninth and it cannot. The structure being forced is arithmetic rather than graphical, and the proof is a different proof.
Solutions that come in multiples of p
Count the solutions of x² + y² + z² = 0 in the field with five elements and there are 25; with seven, there are 49. Whenever a system of equations has more unknowns than its total degree, its number of solutions is a multiple of the characteristic — which forces a solution besides zero, and the reason is a sum over the field that vanishes because its non-zero elements form one cycle.
Every power sum, from the coefficients alone
Raise the roots of a polynomial to the k-th power and add them. However the roots turn, the total is a whole number when the coefficients are, and Newton's identities produce it from the coefficients one step at a time — no root is ever found. Run the rule on x³ − x − 1 and out comes Perrin's sequence, whose terms know which numbers are prime, nearly.
The sum that steps over every whole number
The harmonic sum 1 + 1/2 + 1/3 + … passes 2 at the fourth term, 3 at the eleventh, 4 at the thirty-first, and eventually every whole number there is. It never lands on one. The proof is a single number in the list 1, 2, …, n that carries more factors of two than any other — and the same arithmetic makes the numerators divisible by squares of primes they have no business knowing about.
An order that proves a prime
Fermat's little theorem is a test that primes pass and composites mostly fail, and it can be fooled. Run backwards, it cannot. If some number a has order exactly n − 1 modulo n, then n is prime — because only a prime has n − 1 numbers to cycle through. Checking that takes the prime factors of n − 1, which need proofs of their own, and the proofs nest into a tree that anyone can check: Pratt's certificate, which shows every prime has a short proof of being one.
Two primes where Fermat holds twice
Fermat's theorem says p divides 2^(p−1) − 1. Usually p² does not. It does at 1093 and at 3511 and at no other prime anyone has found, in searches reaching past 10^19. The leftover, (2^(p−1) − 1)/p taken mod p, behaves like a random number, so a prime has about a one-in-p chance of the extra divisibility — and a random count with that chance grows so slowly that two by now is unremarkable, while nobody can prove there are any more, or that there are infinitely many primes where it fails.
Named alongside it
The objects these essays reach for when they reach for this one.
Modular arithmeticCounting argumentCyclic groupPrimality testPrimesCompositeDivisibilityExistence proofFinite fieldModulusOrbitOrder