On the equation $x_1^2+x_2^2+x_3^2+x_4^2=N$ with variables such that $x_1{x}_2{x}_3{x}_4+1$ is an almost-prime

Abstract

We consider Lagrange's equation $x_1^2+x_2^2+x_3^2+x_4^2=N$, where $N$ is a sufficiently large and odd integer, and prove that it has a solution in natural numbers $x_1,\dots ,x_4$ such that $x_1{x}_2{x}_3{x}_4+1$ has no more than 48 prime factors.