Fermat prime
A Fermat prime, named after Pierre de Fermat who first studied them, is a prime number of the form:
- <math> F_{n} = 2^{2^n} + 1 </math>
where n is a natural number. There are only five known Fermat primes: 3 (n=0), 5 (n=1), 17 (n=2), 257 (n=3) and 65537 (n=4). It isn't known whether these are the only Fermat primes, and it isn't even known whether or not there are infinitely many Fermat primes.
Carl Friedrich Gauss proved that there is a relationship between the ruler and compass construction of regular polygons and Fermat primes: a regular n-gon can be constructed with ruler and compasses if and only if n is a power of 2 or the product of a power of 2 and distinct Fermat primes.
Integers of the general form
- <math>2^{2^n}+1 </math>
- <math> F_{5} = 2^{2^5} + 1 = 2^{32} + 1 = 4294967297 = 641 \cdot 6700417 \; </math>
Different Fermat numbers are relatively prime.
See also:
External links:
- Generalized Fermat Prime search (http://perso.wanadoo.fr/yves.gallot/primes/gfn.html)
- http://primes.utm.edu/glossary/page.php?sort=Fermats
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