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Difference between revisions of "Fermat prime"

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(Start article: Fermat prime)
 
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====References====
 
====References====
* Richard K. Guy, ''Unsolved Problems in Number Theory'' 3rd ed. Springer (2004) ISBN 0-387-20860-7 {{ZBL|1058.11001}}
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* 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}}
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* 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}}
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* 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}}

Latest revision as of 19:13, 24 March 2023

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
How to Cite This Entry:
Fermat prime. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fermat_prime&oldid=53156