Fermat numbers

Fermat prime

Named after	Pierre de Fermat
No. of known terms	5
Conjectured no. of terms	5
Subsequence of	Fermat numbers
First terms	3, 5, 17, 257, 65537
Largest known term	65537
OEIS index	A019434

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^k + 1 is prime, and k > 0, it can be shown that k must be a power of two. (If k = ab where 1 ≤ a, b ≤ k and b is odd, then 2^k + 1 = (2^a)^b + 1 ≡ (−1)^b + 1 = 0 (mod 2^a + 1). See below for a complete proof.) In other words, every prime of the form 2^k + 1 (other than 2 = 2⁰ + 1) is a Fermat number, and such primes are called Fermat primes. As of 2016, the only known Fermat primes are F₀, F₁, F₂, F₃, and F₄ (sequence in the OEIS).

The Fermat numbers satisfy the following recurrence relations:

for n ≥ 1,

for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. 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, 2. 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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