Fame, difficulty, and usefulness

Pierre Fermat is best known for two theorems, dubbed his “last” theorem and his “little” theorem. His last theorem is famous, difficult to prove, and useless. His little theorem is relatively arcane, easy to prove, and extremely useful.

There’s little relation between technical difficulty and usefulness.

Fermat’s last theorem

Fermat’s last theorem says there are no positive integer solutions to

an + bn = cn

for integers n > 2. This theorem was proven three and a half centuries after Fermat proposed it. The theorem is well known, even in pop culture. For example, Captain Picard tries to prove it in Star Trek: Next Generation. Fermat’s last theorem was famous for being an open problem that was easy to state. Now that it has been proven, perhaps it will fade from popular consciousness.

The mathematics developed over the years in attempts to prove Fermat’s last theorem has been very useful, but the theorem itself is of no practical importance that I’m aware of.

Fermat’s little theorem

Fermat’s little theorem says that if p is a prime and a is any integer not divisible by p, then ap − 1 − 1 is divisible by p. This theorem is unknown in pop culture but familiar in math circles. It’s proved near the beginning of any introductory number theory class.

The theorem, and its generalization by Euler, comes up constantly in applications of number theory. See three examples here.

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