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 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.