New Results in the Investigation of the Goldbach-Euler Problem and the Problem of Prime Pairs

Goldbach-Euler problem concerning the representation of even number as the sum of two primes has not been solved up to date. With the aid of Eratosthenes sieve method and the theory of Dirichlet L-series developed by Ju. V. Linnik and his successors, one can prove that there exists an integer k such that eyery large number 2N can be represented as 2N = p+n, where p is a prime number and n has at most k prime factors. A. Rényi [1] first established the existence of such k. k = 4 was obtained by B. V. Levin, M. B. Barban, Wang Yuan and Pan Cheng Dong [2,3,4]. The corresponding results are also obtained on the problem of prime pairs, i.e., there exist infinitely many prime numbers p such that p+2 has at most k prime factors. In this paper, I shall prove k = 3…