Concept
Order — where it appears
The number of times an operation has to be repeated before it returns to where it started. It divides the size of the group, which is Lagrange's theorem applied to the subgroup that one element generates.
Named by 2 essays across 2 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.
Named alongside it
The objects these essays reach for when they reach for this one.
CompositeCyclic groupFermats little theoremModular arithmeticModulusOrbitPrimesCounting two waysGreatest common divisorGroup actionParityPeriodicity