On the Equation x2 1 + x2 2 + x2 3 + x2 4 = N with Variables such that x1x2x3x4 + 1 is an Almost-Prime

摘要

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 solutionin natural numbers $x_1, dots, x_4 $ such that $x_1 x_2 x_3 x_4 + 1$ has nomore than 48 prime factors.