THE QUADRATIC FORM IN NINE PRIME VARIABLES

Abstract

Let $f(x_{1},\ldots ,x_{n})$ be a regular indefinite integral quadratic form with $n\geqslant 9$, and let $t$ be an integer. Denote by $\mathbb{U}_{p}$ the set of $p$-adic units in $\mathbb{Z}_{p}$. It is established that $f(x_{1},\ldots ,x_{n})=t$ has solutions in primes if (i) there are positive real solutions, and (ii) there are local solutions in $\mathbb{U}_{p}$ for all prime $p$.