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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 2^{n} + 1 is prime, and n > 0, it can be shown that n must be a power of two. (If n = ab where 1 ≤ a, b ≤ n and b is odd, then 2^{n} + 1 = (2^{a})^{b} + 1 ≡ (−1)^{b} + 1 = 0 (mod 2^{a} + 1). See below for complete proof.) In other words, 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_{1}, F_{2}, F_{3}, and F_{4}.
Contents
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 F_{i} and F_{j} have a common factor a > 1. Then a divides both
and F_{j}; 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 F_{n}, choose a prime factor p_{n}; then the sequence {p_{n}} is an infinite sequence of distinct primes.
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