Sums of squares with restrictions involving primes

Abstract

The well-known Lagrange's four-square theorem states that any integer $n\in\mathbb{N}=\{0,1,2,...\}$ can be written as the sum of four squares. Recently, Z.-W. Sun investigated the representations of $n$ as $x^2+y^2+z^2+w^2$ with certain linear restrictions involving the integer variables $x,y,z,w$. In this paper, via the theory of quadratic forms, we further study the representations $n=x^2+y^2+z^2+w^2$ (resp., $n=x^2+y^2+z^2+2w^2$) with certain linear restrictions involving primes. For example, we obtain the following results: (i) Each positive integer $n>1$ can be written as $x^2+y^2+z^2+2w^2$ ($x,y,z,w\in\mathbb N$) with $x+y$ prime. (ii) Every positive integer can be written as $x^2+y^2+z^2+2w^2$ ($x,y,z,w\in\mathbb N$) with $x+2y$ prime. (iii) Let $k$ be any positive integer, and let $d\ge 2^{k-1}$ be a positive odd integer with $4d^2+1$ prime. Then any sufficiently large integer can be written as $x^2+y^2+z^2+2w^2$ $(x,y,z,w\in\mathbb N)$ with $x+2dy=p^k$ for some prime $p$.