A ternary diophantine inequality by primes near to squares

TL;DR

This paper proves that for large numbers, the inequality involving sums of three prime powers near to squares can be solved, extending understanding of prime distributions in Diophantine inequalities.

Contribution

It establishes the solvability of a specific ternary Diophantine inequality involving primes near squares for large numbers, with fixed exponent c between 1 and 35/34.

Findings

Solvability of the inequality for large N

Primes near to squares can approximate the sum within epsilon

Extension of prime distribution results in Diophantine problems

Abstract

Let $c$ be fixed with $1<c<35/34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} is solvable in primes $p_1,\,p_2,\,p_3$ near to squares.