turkmath.org

Classical Waring-Goldbach problem concerns representability of all large integers satisfying a certain local condition as sums of fixed number of kth powers of prime numbers where k ≥ 1. For instance Goldbach’s conjecture states that every even number ≥ 4 can be expressed as a sum of two primes. Let H(k) be the least integer s such that every sufficiently large positive integer subject to a certain condition may be expressed as a sum of s kth powers of primes. Following the pioneering work of Vinogradov (1937) (which yields H(1) ≤ 3), Hua (1938-1959) showed that H(k) ≤ 2 k + 1. He then reduced his bound to H(k) ≤ 4k log k(1 + o(1)) for every large k. In this talk, we shall first go over the proofs of upper bounds of Hua, and then look at Waring-Goldbach problem with primes chosen from sparse subsets of integers.