A pair of equations in unlike powers of primes and powers of 2

Yuanqiang Cai ,Liqun Hu

doi:10.1515/math-2020-0179

Abstract

Abstract In this article, we show that every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of one prime, one prime squares, two prime cubes, and 187 powers of 2.

Highlights

As an approximation to Goldbach’s problem, Linnik proved in 1951 [1] under the assumption of the Generalized Riemann Hypothesis (GRH), and later in 1953 [2] unconditionally, that each large even integerN is a sum of two primes p1, p2 and a bounded number of powers of 2, namelyN = p1 + p2 + 2ν1 +⋯+ 2νk. (1.1)In 2002, Heath-Brown and Puchta [3] applied a rather different approach to this problem and showed that k = 13 and, on the GRH, k = 7
In an unpublished manuscript, which is yet to appear in print, showed that k = 12; this was proved independently by Liu and Lü [5]
In 1999, Liu et al [6] proved that every large even integer N can be written as a sum of four squares of primes and a bounded number of powers of 2, namely

Summary

Introduction

As an approximation to Goldbach’s problem, Linnik proved in 1951 [1] under the assumption of the Generalized Riemann Hypothesis (GRH), and later in 1953 [2] unconditionally, that each large even integer. In 1999, Liu et al [6] proved that every large even integer N can be written as a sum of four squares of primes and a bounded number of powers of 2, namely. In 2001, Liu and Liu [8] proved that every large even integer N can be written as a sum of eight cubes of primes and a bounded number of powers of 2, namely. The acceptable value k = 330 was determined by Platt and Trudgian [7]. N = p1 + p22 + p33 + p43 + 2v1 +⋯+ 2vk They showed that k = 161 is acceptable and Platt and Trudgian [7] revised it to 156

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Let q

So we get