How did Fermat discover his theorem?

Abstract

In 1640 Pierre de Fermat discovered his theorem that if $p$ is prime and $a$ is not divisible by $p$, then $a^{p-1}-1$ is divisible by $p$; or, as we write today, $a^{p-1}\equiv1\pmod{p}$. This is perhaps the first and the most important surprising property ever discovered about primes. There is little in number theory that is not dependent on it or intertwined with it, and its significance is amply demonstrated by the fact that today, almost four centuries later, Fermat's theorem provides the mathematical foundation for the RSA cryptosystem, which is still central to society's communications security even after several decades serving as its heart. Fermat's theorem is totally unexpected and truly astonishing. So why and how did he discover it? We know that Fermat was studying perfect numbers from classical Greek mathematics. But exactly how did that lead to his discovery? The secret lies in patterns in prime factorizations of Mersenne numbers, and Fermat's letters reveal hints of his path. We can reconstruct details of how Mersenne numbers led to Fermat's discoveries.