The quadratic form in 9 prime variables

TL;DR

This paper proves that indefinite quadratic forms in at least nine prime variables have solutions for any integer value, provided there are no local obstructions, extending the understanding of prime solutions in quadratic equations.

Contribution

It establishes the existence of prime solutions for indefinite quadratic forms in nine or more variables under local conditions, advancing previous results in number theory.

Findings

Solutions exist in prime variables for indefinite quadratic forms with n≥9

No local obstructions are necessary for the existence of solutions

The result applies to all integers t without additional restrictions

Abstract

Let $f(x_1,\ldots,x_n)$ be a regular indefinite integral quadratic form with $n\ge 9$, and let $t$ be an integer. It is established that $f(x_1,\ldots,x_n)=t$ has solutions in prime variables if there are no local obstructions.