Divisor
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
Which roots refuse to be fractions
The square root of two is not a fraction, and neither is the square root of three, five, six or seven. The rule behind the list turns an infinite question into a search over the divisors of a single number — and the search finishes.
One residue whose powers are all of them
Fermat's theorem says every order divides p − 1. It does not say that anything has order exactly p − 1, which is a separate and stronger claim — and what forces it is a count of how many numbers share each divisor with p − 1.
The exponent that is smaller than Euler's
Euler's theorem raises every unit to the count of the units and gets one. The smallest exponent that works for all of them at once is often much smaller — and a composite is invisible to Fermat's test exactly when that smaller number divides n − 1.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting two waysModular arithmeticOrderPrimesAlgebraic numberCompositeCyclic groupExhaustive searchGroup actionInfinite descentIntegralityIrrationality