Prime Solutions of Diagonal Diophantine Systems

TL;DR

This paper establishes an asymptotic formula for counting prime solutions to diagonal Diophantine systems, connecting advanced mean value bounds and local solvability, with implications for Vinogradov systems and the Waring-Goldbach problem.

Contribution

It provides a new asymptotic formula for prime solutions of diagonal systems, combining mean value bounds, local solvability, and existing theorems, advancing understanding of prime solutions in Diophantine equations.

Findings

Asymptotic formula for prime solutions of diagonal systems

Conditional results for Vinogradov systems

Implications for Waring-Goldbach problem on seven cubes

Abstract

An asymptotic formula for the number of prime solutions of a general diagonal system of Diophantine equations is established, contingent on the existence of an appropriate mean value bound and on local solvability. In conjunction with the Vinogradov Mean Value Theorem this yields an asymptotic formula for solutions of Vinogradov systems and in conjunction with Hooley's work on seven cubes this yields a conditional result for the Waring-Goldbach problem on seven cubes of primes, contingent on Hooley's form of the Riemann hypothesis.