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Fermat number
A Fermat number is a special positive number. Fermat numbers are named after Pierre de Fermat. The formula that generates them is
 [math]F_{n} = 2^{2^{ \overset{n} {}}} + 1[/math]
where n is a nonnegative integer. The first nine Fermat numbers are (sequence A000215 in OEIS):
 F_{0} = 2^{1} + 1 = 3
 F_{1} = 2^{2} + 1 = 5
 F_{2} = 2^{4} + 1 = 17
 F_{3} = 2^{8} + 1 = 257
 F_{4} = 2^{16} + 1 = 65537
 F_{5} = 2^{32} + 1 = 4294967297 = 641 × 6700417
 F_{6} = 2^{64} + 1 = 18446744073709551617 = 274177 × 67280421310721
 F_{7} = 2^{128} + 1 = 340282366920938463463374607431768211457 = 59649589127497217 × 5704689200685129054721
 F_{8} = 2^{256} + 1 = 115792089237316195423570985008687907853269984665640564039457584007913129639937 = 1238926361552897 × 93461639715357977769163558199606896584051237541638188580280321
As of 2007, only the first 12 Fermat numbers have been completely factored. (written as a product of prime numbers) These factorizations can be found at Prime Factors of Fermat Numbers.
If 2^{n} + 1 is prime, and n > 0, it can be shown that n must be a power of two. Every prime of the form 2^{n} + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F_{0},...,F_{4}.
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Interesting things about Fermat numbers
 No two Fermat numbers have common divisors.
 Fermat numbers can be calculated recursively: To get the Nth number, multiply all Fermat numbers before it, and add two to the result.
What they are used for
Today, Fermat numbers can be used to generate random numbers, between 0 and some value N, which is a power of 2.
Fermat's conjecture
Fermat, when he was studying these numbers, conjectured that all Fermat numbers were prime. This was proven to be wrong by Leonhard Euler, who factorised [math]F_5[/math] in 1732.
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