Fermat number

Fermat numbers are a sequence of numbers of the form 2^(2^n) + 1, where n is a non-negative integer. They are named after the mathematician Pierre de Fermat, who first studied them in the 17th century.

The first few Fermat numbers are:

F(0) = 2^(2^0) + 1 = 3
F(1) = 2^(2^1) + 1 = 5
F(2) = 2^(2^2) + 1 = 17
F(3) = 2^(2^3) + 1 = 257
F(4) = 2^(2^4) + 1 = 65537

Fermat numbers have many interesting properties and connections to other areas of mathematics. One of the most famous is Fermat's Little Theorem, which states that if p is a prime number, then for any integer a not divisible by p, a^(p-1) - 1 is divisible by p. This theorem can be used to prove that if n is a non-negative integer and 2^(2^n) + 1 is prime, then n must be a power of 2.

However, not all Fermat numbers are prime. In fact, F(5) is known to be composite, and it is not known whether any of the other Fermat numbers beyond F(5) are prime. It is conjectured that there are infinitely many prime Fermat numbers, but this has not been proven. The search for such primes has led to the discovery of several large prime numbers, including the current record holder for the largest known prime, which is a Mersenne prime (a prime number of the form 2^p - 1) with over 24 million digits.