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 Template: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.