Fermat number - Simple English Wikipedia, the free encyclopedia

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

${\displaystyle F_{n}=2^{2^{\overset {n}{}}}+1}$

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

F₀ = 2¹ + 1 = 3

F₁ = 2² + 1 = 5

F₂ = 2⁴ + 1 = 17

F₃ = 2⁸ + 1 = 257

F₄ = 2¹⁶ + 1 = 65537

F₅ = 2³² + 1 = 4294967297 = 641 × 6700417

F₆ = 2⁶⁴ + 1 = 18446744073709551617 = 274177 × 67280421310721

F₇ = 2¹²⁸ + 1 = 340282366920938463463374607431768211457 = 59649589127497217 × 5704689200685129054721

F₈ = 2²⁵⁶ + 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 2ⁿ + 1 is prime, and n > 0, it can be shown that n must be a power of two. Every prime of the form 2ⁿ + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F₀,...,F₄.

• 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.

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

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 ${\displaystyle F_{5}}$ in 1732.

• Sequence of Fermat numbers Archived 2001-07-16 at the Wayback Machine
• Prime Glossary Page on Fermat Numbers
• Generalized Fermat Prime search
• History of Fermat Numbers Archived 2007-09-28 at the Wayback Machine
• Unification of Mersenne and Fermat Numbers Archived 2006-10-02 at the Wayback Machine
• Prime Factors of Fermat Numbers Archived 2016-02-10 at the Wayback Machine
• Fermat Number at MathWorld
• Distributed Search for Fermat Number Divisors