On Two Diophantine Inequalities Over Primes

TL;DR

This paper proves that for almost all large numbers, certain Diophantine inequalities involving sums of prime powers with exponent c are solvable in primes, extending results to three and six primes.

Contribution

It establishes solvability of specific Diophantine inequalities over primes for a range of c and large N, including cases with three and six primes, which is a novel extension.

Findings

Almost all R in (N, 2N] satisfy the inequality with three primes.

The six-prime inequality is solvable for sufficiently large N.

The results cover c in (1, 37/18) excluding 2.

Abstract

Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$ Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3,\,p_4,\,p_5,\,p_6$ for sufficiently large real number $N$.