On a Diophantine equation with five prime variables

TL;DR

This paper proves that for a specific range of real numbers c, large integers N can be expressed as the sum of the integral parts of five prime powers, advancing understanding of prime-based Diophantine equations.

Contribution

It establishes the solvability of a particular five-prime-variable Diophantine equation for a range of c values, extending previous results in prime number theory.

Findings

The equation is solvable for large N when 1<c<4109054/1999527, c≠2.

Prime variables p_i exist such that the sum of their c-th powers' integer parts equals N.

The result applies to a broad class of exponents c, excluding c=2.

Abstract

Let $[x]$ denote the integral part of the real number $x$, and $N$ be a sufficiently large integer. In this paper, it is proved that, for $1<c<\frac{4109054}{1999527}, c\not=2$, the Diophantine equation $N=[p_1^c]+[p_2^c]+[p_3^c]+[p_4^c]+[p_5^c]$ is solvable in prime variables $p_1,p_2,p_3,p_4,p_5$.