On two Diophantine inequalities over primes (II)

TL;DR

This paper proves that certain Diophantine inequalities involving prime variables and fractional powers are solvable for almost all large numbers, extending previous bounds on the exponent c.

Contribution

It establishes new solvability results for Diophantine inequalities over primes with a broader range of the exponent c, improving prior bounds.

Findings

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

The six-prime inequality is solvable for a wider range of c

Improves previous bounds on the exponent c in such inequalities

Abstract

Let $1<c<\frac{26088036}{12301745},c\not=2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N \end{equation*} is solvable in primes $p_1,p_2,p_3$. Moreover, we also prove that the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4,p_5,p_6$, which improves the previous result $1<c<\frac{37}{18},c\neq2$.