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 nonnegative integer. The first few Fermat numbers are:
- 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, … (sequence A000215 in OEIS).
If 2n + 1 is prime, and n > 0, it can be shown that n must be a power of two. (If n = ab where 1 ≤ a, b ≤ n and b is odd, then 2n + 1 = (2a)b + 1 ≡ (−1)b + 1 = 0 (mod 2a + 1). See Sec. 5 for complete proof.) In other words, every prime of the form 2n + 1 is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F0, F1, F2, F3, and F4.
Read more about Fermat Number: Basic Properties, Primality of Fermat Numbers, Factorization of Fermat Numbers, Pseudoprimes and Fermat Numbers, Other Theorems About Fermat Numbers, Relationship To Constructible Polygons, Other Interesting Facts, Generalized Fermat Numbers
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