Waring–Goldbach problem: two squares and three biquadrates

Abstract

Let ℛ(n) denote the number of representations of the positive integer n as the sum of two squares and three biquadrates of primes and we write ℰ(N) for the number of positive integers n satisfying n≤N, n≡5,53,101(mod120) and | ℛ ( n ) − Γ 2 ( 1 2 ) Γ 3 ( 1 4 ) Γ ( 7 4 ) 𝔖 ( n ) n 3 4 log 5 n | ≥ n 3 4 log 1 1 2 n , where 0<𝔖(n)≪1 is the singular series. In this paper, we prove ℰ ( N ) ≪ N 1 5 3 2 + 𝜀 for any 𝜀>0. This result constitutes a refinement upon that of Friedlander and Wooley (2014).