Fermat number

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In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form

where n is a nonnegative integer. The first few Fermat numbers are:

If 2n + 1 is prime, and n > 0, it can be shown that n must be a power of two. (If n = ab where 1 ≤ a, bn and b is odd, then 2n + 1 = (2a)b + 1 ≡ (−1)b + 1 = 0 (mod 2a + 1). See below for complete proof.) In other words, every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F0, F1, F2, F3, and F4.


Basic properties

The Fermat numbers satisfy the following recurrence relations

for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem: no two Fermat numbers share a common factor. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > 1. Then a divides both

and Fj; hence a divides their difference too. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each Fn, choose a prime factor pn; then the sequence {pn} is an infinite sequence of distinct primes.

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