On the exceptional set of Hardy-Littlewood's numbers in short intervals

Abstract

In 1923 Hardy and Littlewood [4] conjectured that every sufficiently large integer is either a k-power of an integer or a sum of a prime and a k-power of an integer, for k = 2, 3. Define an Hardy-Littlewood number (HL-number) to be an integer which is a sum of a prime and of a k-power of an integer, k ∈ N, k ≥ 2. Let X be a sufficiently large parameter. Denote by Ek the set of integers which are neither an HL-number nor a k-power of an integer, let Ek(X) = Ek ∩ [1, X] and Ek(X,H) = Ek ∩ [X,X +H], where H = o(X). Hardy-Littlewood’s conjectures are equivalent to Ek(X) 1. The best known result on E2(X) was independently proved by Brunner-Perelli-Pintz [1] and A.I. Vinogradov [14]: