Order of an element
Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
Differences round a ring
Write four numbers at the corners of a square and put on each side the difference of its two corners. Repeat with the new square. Whatever the four whole numbers are, the process reaches nought within a few steps. Write five numbers and it almost never does. Rings whose size is a power of two always finish, and every other ring settles into a cycle whose length depends on how a polynomial factors over two elements: 7 at seven, 341 at eleven, 31 at thirty-one.
Numbers that never make a prime
Multiply 78,557 by a power of two and add one. The result is never prime, for any power, though almost every other odd number gives a prime within a few doublings. The reason is that seven small primes take it in turn to divide the result, on a schedule repeating every 36 steps that leaves no step uncovered. Proving nothing smaller does this has taken sixty years of search, and five candidates are still running.
Named alongside it
The objects these essays reach for when they reach for this one.
Fermats little theoremConjectureExhaustive searchPrimality testPrimePrimesCarmichael numberCongruenceContinued fractionsCoveringDensityFermat last theorem