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
Famous quotes containing the word number:
“Envy has blackened every page of his history.... The future, in its justice, will number him among those men whom passions and an excess of activity have condemned to unhappiness, through the gift of genius.”
—Eugène Delacroix (1798–1863)