What is a Fermat number?
Q: What is a Fermat number?
A: A Fermat number is a special positive number named after Pierre de Fermat. It is generated by the formula F_n = 2^2^(n) + 1, where n is a nonnegative integer.
Q: How many Fermat numbers are there?
A: As of 2007, only the first 12 Fermat numbers have been completely factored.
Q: What are the first nine Fermat numbers?
A: The first nine Fermat numbers are F0 = 3, F1 = 5, F2 = 17, F3 = 257, F4 = 65537, F5 = 4294967297 (641 × 6700417), F6 = 18446744073709551617 (274177 × 67280421310721), F7 = 340282366920938463463374607431768211457 (59649589127497217 × 5704689200685129054721), and F8 = 115792089237316195423570985008687907853269984665640564039457584007913129639937 (1238926361552897 × 93461639715357977769163558199606896584051237541638188580280321).
Q: What can be said about prime numbers of the form 2n + 1?
A: If 2n + 1 is prime and n > 0 then it can be shown that n must be a power of two. Every prime of the form 2n + 1 is also a Fermat number and such primes are called Fermat primes. The only known Fermat primes are from 0 to 4.
Q: Where can one find factorizations for all 12 known factoredFermat numbers?
A: Factorizations for all 12 known factoredFermat numbers can be found at Prime Factors of Fermat Numbers.
Q: Who was Pierre de Fermaat?
A: Pierre de Fermaat was an influential French mathematician who lived in the 17th century and whose work laid much of the groundwork for modern mathematics. He is best known for his contributions to probability theory and analytic geometry as well as his famous Last Theorem which remained unsolved until 1995 when it was finally proven by Andrew Wiles using methods from algebraic geometry.
