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What (who) is Fermat primes - definition
POSITIVE INTEGER OF THE FORM (2^(2^N))+1
Fermat prime; Fermat numbers; 4294967297 (number); Fermat primes; Fermat Numbers; Generalized Fermat number; Generalized Fermat prime; Fermat Primes; Fermat Prime; 4294967297; Primality of Fermat numbers; Factorization of Fermat numbers; Generalized Fermat numbers; Generalized Fermat primes; Generalized Fermat
Fermat prime
<mathematics> A prime number of the form 2^2^n + 1. Any
prime number of the form 2^n+1 must be a Fermat prime.
Fermat conjectured in a letter to someone or other that all
numbers 2^2^n+1 are prime, having noticed that this is true
for n=0,1,2,3,4.
Euler proved that 641 is a factor of 2^2^5+1. Of course
nowadays we would just ask a computer, but at the time it was
an impressive achievement (and his proof is very elegant).
No further Fermat primes are known; several have been
factorised, and several more have been proved composite
without finding explicit factorisations.
Gauss proved that a regular N-sided polygon can be
constructed with ruler and compasses if and only if N is a
power of 2 times a product of distinct Fermat primes.
(1995-04-10)
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
Fermat curve
MATHEMATICAL CONCEPT
Fermat varieties; Fermat variety
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation
Wikipedia
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
where n is a non-negative integer. The first few Fermat numbers are:
- 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... (sequence A000215 in the OEIS).
If 2k + 1 is prime and k > 0, then k must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in the OEIS); heuristics suggest that there are no more.
