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# Fermat number

A **Fermat number** is a special positive number. Fermat numbers are named after Pierre de Fermat. The formula that generates them is

- [math]F_{n} = 2^{2^{ \overset{n} {}}} + 1[/math]

where *n* is a nonnegative integer. The first nine Fermat numbers are (sequence A000215 in OEIS):

*F*_{0}= 2^{1}+ 1 = 3*F*_{1}= 2^{2}+ 1 = 5*F*_{2}= 2^{4}+ 1 = 17*F*_{3}= 2^{8}+ 1 = 257*F*_{4}= 2^{16}+ 1 = 65537*F*_{5}= 2^{32}+ 1 = 4294967297 = 641 × 6700417*F*_{6}= 2^{64}+ 1 = 18446744073709551617 = 274177 × 67280421310721*F*_{7}= 2^{128}+ 1 = 340282366920938463463374607431768211457 = 59649589127497217 × 5704689200685129054721*F*_{8}= 2^{256}+ 1 = 115792089237316195423570985008687907853269984665640564039457584007913129639937 = 1238926361552897 × 93461639715357977769163558199606896584051237541638188580280321

As of 2007, only the first 12 Fermat numbers have been completely factored. (written as a product of prime numbers) These factorizations can be found at Prime Factors of Fermat Numbers.

If 2^{n} + 1 is prime, and *n* > 0, it can be shown that *n* must be a power of two. 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*_{4}.

## Contents

## Interesting things about Fermat numbers

- No two Fermat numbers have common divisors.
- Fermat numbers can be calculated recursively: To get the Nth number, multiply all Fermat numbers before it, and add two to the result.

## What they are used for

Today, Fermat numbers can be used to generate random numbers, between 0 and some value N, which is a power of 2.

## Fermat's conjecture

Fermat, when he was studying these numbers, conjectured that all Fermat numbers were prime. This was proven to be wrong by Leonhard Euler, who factorised [math]F_5[/math] in 1732.