A Fermat prime is one of the form $2^{2^n}+1.$ Pierre de Fermat once conjectured that all numbers of this form are prime, as indeed the first few are: 3, 5, 17, 257, 65537. Unfortunately, it turns out that $2^{2^5}+1=2^{32}+1$ is not prime; it is a multiple of 641, as Leonhard Euler discovered. It's now thought likely that no Fermat numbers after 65537 are prime.

A regular polygon with n sides is constructible with ruler and compass if and only if n is the product of 0 or more distinct Fermat primes, and a power of 2.

Exercise: prove that if $2^m+1$ is prime, then m is a power of 2. (Hint: if m is not a power of 2 then it has an odd factor other than 1. Can you use this to find a factor of $2^m+1$ ?)

Another exercise: prove that $2^{32}+1$ is a multiple of 641 without doing any nontrivial arithmetic, using the fact that $641=5.2^7+1=5^4+2^4.$

See also: Mersenne prime.

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