Type: Article
Publication Date: 2002-01-01
Citations: 80
DOI: https://doi.org/10.4310/ajm.2002.v6.n3.a7
Introduction.It was shown by Linnik [10] that there is an absolute constant K such that every sufficiently large even integer can be written as a sum of two primes and at most K powers of two.This is a remarkably strong approximation to the Goldbach Conjecture.It gives us a very explicit set IC(x) of integers n < x of cardinality only 0((logx) K ), such that every sufficiently large even integer N < x can be written as N = p + p' + n, with p,p' prime and n G /C(x).In contrast, if one tries to arrange such a representation using an interval in place of the set /C(a:), all known results would require lC(x) to have cardinality at least a positive power of x.Linnik did not establish an explicit value for the number K of powers of 2 that would be necessary in his result.However, such a value has been computed by Liu, Liu and Wang [12], who found that K = 54000 is acceptable.This result was subsequently improved, firstly by Li [8] who obtained K = 25000, then by Wang [18], who found that K = 2250 is acceptable, and finally by Li [9] who gave the value K = 1906.One can do better if one assumes the Generalized Riemann Hypothesis, and Liu, Liu and Wang [13] showed that K = 200 is then admissible.The object of this paper is to give a rather different approach to this problem, which leads to dramatically improved bounds on the number of powers of 2 that are required for Linnik's theorem.THEOREM 1.Every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2.