Edit made on August 13, 2011 by ColinWright at 22:43:51
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A Fermat prime is one of the form EQN: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 EQN: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.
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Exercise: prove that if EQN: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 EQN:2^m+1 ?)
Another exercise: prove that EQN:2^32+1 EQN:2^{32}+1 is a multiple of 641 without doing any
nontrivial arithmetic, using the fact that EQN:641=5.2^7+1=5^4+2^4.
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See also: Mersenne prime.