View source for Fermat prime

{{TEX|done}}{{MSC|11A51}}

A [[prime number]] of the form $F_k = 2^{2^k}+1$ for a natural number $k$.  They are named after Pierre de Fermat who observed that $F_0,F_1,F_2,F_3,F_4$ are prime and that this sequence "might be indefinitely extended".  To date (2021), no other prime of this form has been found, and it is known, for example, that $F_k$ is composite for $k=5,\ldots,32$.  Lucas has given an efficient test for the primality of $F_k$.  The Fermat primes are precisely those odd primes $p$ for which a ruler-and-compass construction of the regular $p$-gon is possible: see [[Geometric constructions]] and [[Cyclotomic polynomials]].


====References====
* Richard K. Guy, ''Unsolved Problems in Number Theory'' 3rd ed. Springer (2004) ISBN 0-387-20860-7 {{ZBL|1058.11001}}
* G.H. Hardy; E.M. Wright. "An Introduction to the Theory of Numbers", Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press (2008) [1938] ISBN 978-0-19-921986-5 {{ZBL|1159.11001}}
* Michal Krizek, Florian Luca, Lawrence Somer, "17 Lectures on Fermat Numbers: From Number Theory to Geometry" Springer (2001) ISBN 0-387-21850-5 {{ZBL|1010.11002}}