The Number Of Powers Of 2 In A Representation Of Large Odd Integers

In 1742. Goldbach formulated the conjecture that every even integer not less than 6 is a sum of two odd primes. The conjecture still remains open. In 1923. Hardy and Littlewood proved that assuming GRH,where E(x) is the number of positive even integers not exceeding x which can not be represented as sums of two primes,e > 0 is arbitrary and << is the Vinogradov symbol.In 1951 and 1953, Linnik established then following ''almost Goldbach" result that each large even integer N is a sum of two primes p₁,p₂ and a bounded number of powers of 2.where (and throughout) p and v, with or without subscripts, denote a prime number and a positive integer respectively. This result was generalized by Vinogradov in several directions. In 1975. Gallagber considerably-simplified the proofs of Linnik and Vinogradov. and established the following : For any integerk ≥ 2. there is N_k > 0 depending on k only such that for each even integer N ≥N_k,where r_k″(N) is the number of representations of N in the form of (2)An explicit value for the number k of powers of 2 was firstly establish by Liu and Wang [2], who found that k = 54000 is acceptable. The original value for the number k was subsequently improved by Li [4], Wang [23]. and Li[5]. Recently Heath-Brown and Puchta [21] applied a rather different approach to this problem and showed that k = 13 is acceptable.