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 + 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 + 1 = (2) + 1 ≡ (−1) + 1 = 0 (mod 2 + 1). See below for a complete proof.) In other words, every prime of the form 2 + 1 (other than 2 = 2 + 1) is a Fermat number, and such primes are called Fermat primes. As of 2015, the only known Fermat primes are F₀, F₁, F₂, F₃, and F₄ (sequence A019434 in OEIS).