Chen’s conjecture and its generalization

Abstract

Let l1, l2, ..., lg be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that (k + l1)2, (k + l2)2, ..., (k + lg)2 can not be expressed as 2n + pα is at least c(g)x, where p is an odd prime and the constant c(g) depends only on g.