Fermat number

In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form

Related Topics:
Mathematics - Pierre de Fermat - Positive integer

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:F_{n} = 2^{2^n} + 1

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where n is a nonnegative integer. The first eight Fermat numbers are {{OEIS|id=A000215}}:

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:F0 = 21 + 1 = 3

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:F1 = 22 + 1 = 5

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:F2 = 24 + 1 = 17

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:F3 = 28 + 1 = 257

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:F4 = 216 + 1 = 65537

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:F5 = 232 + 1 = 4294967297 = 641 × 6700417

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:F6 = 264 + 1 = 18446744073709551617 = 274177 × 67280421310721

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:F7 = 2128 + 1 = 340282366920938463463374607431768211457 = 59649589127497217 × 5704689200685129054721

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Only the first 12 Fermat numbers have been completely factorised. These factorisations can be found at Prime Factors of Fermat Numbers

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If 2n + 1 is prime, it can be shown that n must be a power of 2. (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).) 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,...,F4.

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