The number of 8n+1 primes compare with 8n-1 primes

Abstract

There is a phenomenon in mathematics: there is a phenomenon that before a natural number K, primes of the form 4n+1 do not appear more frequently than 4n-1 primes; Beyond k, between k and k+k', the above phenomenon is reversed, the frequency of 4n+1 primes is not less than 4n-1 primes; After exceeding k+k', between k+k 'and k+k'+k'', it is reversed again ...... The J.E. Littlewood proved the first stage of the phenomenon: primes of the form 4n+1 appear no more frequently than 4n-1 primes before a natural number k. In this paper used a more easy method and directly prove the phenomenon very shortly , provides a theoretical proof for this description.This method is more easy directly and elementary than Littlewood’,and It can help people understand this phenomenon better, and at the same time, it provides a good example for the optimization of number theory research methods and the use of some elementary methods to study mathematical problems. At the same time, there is a generalization conjecture: before a natural number K, which of 8n+1 and 8N-1 primes appear more frequently? The conjecture remains unsolved. Littlewood proved the occurrence frequency theorem of 4n+1 primes and 4N-1 primes, and this paper also gave the proof, the method is different from Littlewood, but he was the first; However, for 8n+1 primes compare with 8n-1 primes, we prove for the first time that the result is same as 4n+1 primes compare with 4n-1 primes.