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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 second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.
When p is a prime, p 2 is a Fermat pseudoprime to base b if and only if p is a Wieferich prime to base b. For example, 1093 2 = 1194649 is a Fermat pseudoprime to base 2, and 11 2 = 121 is a Fermat pseudoprime to base 3. The number of the values of b for n are (For n prime, the number of the values of b must be n − 1, since all b satisfy the ...
For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors. While Carmichael numbers are substantially rarer than prime numbers (Erdös' upper bound for the number of Carmichael numbers [ 3 ] is lower than the prime number function n/log(n) ) there are enough of them that Fermat ...
For integer b > 1, base b may be used if and only if only a finite number of Fermat numbers F n satisfies that () =, where () is the Jacobi symbol. In fact, Pépin's test is the same as the Euler-Jacobi test for Fermat numbers, since the Jacobi symbol ( b F n ) {\displaystyle \left({\frac {b}{F_{n}}}\right)} is −1, i.e. there are no Fermat ...
Cunningham numbers of the form b n − 1 can only be prime if b = 2 and n is prime, assuming that n ≥ 2; these are the Mersenne numbers. Numbers of the form b n + 1 can only be prime if b is even and n is a power of 2, again assuming n ≥ 2; these are the generalized Fermat numbers, which are Fermat numbers when b = 2.
Let A 1 be the set whose elements are the numbers b 1, ab 1, a 2 b 1, ..., a k − 1 b 1 reduced modulo p. Then A 1 has k distinct elements because otherwise there would be two distinct numbers m, n ∈ {0, 1, ..., k − 1} such that a m b 1 ≡ a n b 1 (mod p), which is impossible, since it would follow that a m ≡ a n (mod p).
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...
The five known Fermat primes are: F 0 = 3, F 1 = 5, F 2 = 17, F 3 = 257, and F 4 = 65537 (sequence A019434 in the OEIS). Since there are 31 nonempty subsets of the five known Fermat primes, there are 31 known constructible polygons with an odd number of sides. The next twenty-eight Fermat numbers, F 5 through F 32, are known to be composite. [3]
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