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

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

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]\displaystyle{ F_5 }[/math] in 1732.

Fermat Number Media

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  Number of sides of known constructible polygons having up to 1000 sides (bold) or odd side count (red)

Other websites

• 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