Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Fermat numbers∗ drini† 2013-03-21 12:19:17 The n-th Fermat number is defined as: n Fn = 22 + 1. Fermat incorrectly conjectured that all these numbers were primes, although he had no proof. The first 5 Fermat numbers: 3, 5, 17, 257, 65537 (corresponding to n = 0, 1, 2, 3, 4) are all primes (so called Fermat primes) Euler was the first to point out the falsity of Fermat’s conjecture by proving that 641 is a divisor of F5 . (In fact, F5 = 641 × 6700417). Moreover, no other Fermat number is known to be prime for n > 4, so now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many composite Fermat numbers or not. One of the famous achievements of Gauss was to prove that the regular polygon of m sides can be constructed with ruler and compass if and only if m can be written as m = 2k Fr1 Fr2 · · · Frt where k ≥ 0 and the other factors are distinct primes of the form Fn (of course, t may be 0 here, i.e. m = 2k is allowed). There are many interesting properties involving Fermat numbers. For instance: Fm = F0 F1 · · · Fm−1 + 2 for any m ≥ 1, which implies that Fm − 2 is divisible by all smaller Fermat numbers. The previous formula holds because m m m−1 Fm − 2 = (22 + 1) − 2 = 22 − 1 = (22 − 1)(22 m−1 m−1 + 1) = (22 − 1)Fm−1 and expanding recursively the left factor in the last expression gives the desired result. ∗ hFermatNumbersi created: h2013-03-21i by: hdrinii version: h30068i Privacy setting: h1i hDefinitioni h81T45i h81T13i h11A51i h20L05i h46L87i h43A35i h43A25i h22D25i h55U40i h18B40i h46L05i h22A22i h81R50i h55U35i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 References. Krı́zek, Luca, Somer. 17 Lectures on Fermat Numbers. CMS Books in Mathematics. 2