Fermat prime

2020 Mathematics Subject Classification: Primary: 11A51 [MSN][ZBL]

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