Authors: Pingyuan Zhou
It is well known that there are infinitely many prime factors of Fermat numbers, because prime factor of a Fermat prime is the Fermat prime itself but a composite Fermat number has at least two prime factors and Fermat numbers are pairwise relatively prime. Hence we conjecture that there is at least one prime factor (k^(1/2)*2^(a/2))^2+1 of Fermat number for F(n)-1≤a<F(n+1)-1 (n=0,1,2,3,…), where k^(1/2)is odd posotive integer, a is even positive integer and F(n) is Fermat number. The conjecture holds till a<F(4+1)-1=4294967296 from known evidences. Two corollaries of the conjecture imply existence of infinitely many primes of the form x^2+1, which is one of four basic problems about primes mentioned by Landau at ICM 1912.
Comments: 5 Pages. Auther presents a conjecture related to distribution of a kind of special prime factors of Fermat numbers, which may imply existence of infinitely many primes of the form x^2+1.
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[v1] 2014-06-30 01:00:00
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