On the equation $x^2_1 + x^2_2 + x^2_3 + x^2_4 = N$ with variables such that $x_1 x2_ x_3 x_4 + 1$ is an almost-prime

Abstract

We consider Lagrange's equation

$x^2_1 +x^2_2 +x^2_3 +x^2_4 = N$,

where $ N$ is a sucientlylarge and odd integer, and prove that it has a solution

in natural numbers x1, \ldot,  x4
such that $x_1 x_2 x_3 x_4 + 1$ has no more than 48 prime factors.