Fermat number

In mathematics a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the formLemma. — If n is a positive integer, ( a − b ) ∑ k = 0 n − 1 a k b n − 1 − k = ∑ k = 0 n − 1 a k + 1 b n − 1 − k − ∑ k = 0 n − 1 a k b n − k = a n + ∑ k = 1 n − 1 a k b n − k − ∑ k = 1 n − 1 a k b n − k − b n = a n − b n {displaystyle {egin{aligned}(a-b)sum _{k=0}^{n-1}a^{k}b^{n-1-k}&=sum _{k=0}^{n-1}a^{k+1}b^{n-1-k}-sum _{k=0}^{n-1}a^{k}b^{n-k}\&=a^{n}+sum _{k=1}^{n-1}a^{k}b^{n-k}-sum _{k=1}^{n-1}a^{k}b^{n-k}-b^{n}\&=a^{n}-b^{n}end{aligned}}} Theorem —  If 2 k + 1 {displaystyle 2^{k}+1} is an odd prime, then k {displaystyle k} is a power of 2.If k {displaystyle k} is a positive integer but not a power of 2, it must have an odd prime factor s > 2 {displaystyle s>2} , and we may write k = r s {displaystyle k=rs} where 1 ≤ r < k {displaystyle 1leq r<k} .Theorem —  A Fermat prime cannot be a Wieferich prime.We show if p = 2 m + 1 {displaystyle p=2^{m}+1} is a Fermat prime (and hence by the above, m is a power of 2), then the congruence 2 p − 1 ≡ 1 mod p 2 {displaystyle 2^{p-1}equiv 1{mod {p^{2}}}} does not hold.Theorem (Édouard Lucas) —  Any prime divisor p of F n = 2 2 n + 1 {displaystyle F_{n}=2^{2^{n}}+1} is of the form k 2 n + 2 + 1 {displaystyle k2^{n+2}+1} whenever n > 1.Let Gp denote the group of non-zero elements of the integers modulo p under multiplication, which has order p-1. Notice that 2 (strictly speaking, its image modulo p) has multiplicative order dividing 2 n + 1 {displaystyle 2^{n+1}} in Gp (since 2 2 n + 1 {displaystyle 2^{2^{n+1}}} is the square of 2 2 n {displaystyle 2^{2^{n}}} which is −1 modulo Fn), so that, by Lagrange's theorem, p − 1 is divisible by 2 n + 1 {displaystyle 2^{n+1}} and p has the form k 2 n + 1 + 1 {displaystyle k2^{n+1}+1} for some integer k, as Euler knew. Édouard Lucas went further. Since n > 1, the prime p above is congruent to 1 modulo 8. Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue modulo p, that is, there is integer a such that p | a 2 − 2. {displaystyle p|a^{2}-2.} Then the image of a has order 2 n + 2 {displaystyle 2^{n+2}} in the group Gp and (using Lagrange's theorem again), p − 1 is divisible by 2 n + 2 {displaystyle 2^{n+2}} and p has the form s 2 n + 2 + 1 {displaystyle s2^{n+2}+1} for some integer s. 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: If 2k + 1 is prime, and k > 0, it can be shown that k must be a power of two. (If k = ab where 1 ≤ a, b ≤ k and b is odd, then 2k + 1 = (2a)b + 1 ≡ (−1)b + 1 = 0 (mod 2a + 1). See below for a complete proof.) In other words, every prime of the form 2k + 1 (other than 2 = 20 + 1) is a Fermat number, and such primes are called Fermat primes. As of 2019, the only known Fermat primes are F0, F1, F2, F3, and F4 (sequence A019434 in the OEIS). The Fermat numbers satisfy the following recurrence relations: for n ≥ 1, for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > 1. Then a divides both and Fj; hence a divides their difference, 2. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each Fn, choose a prime factor pn; then the sequence {pn} is an infinite sequence of distinct primes.