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

TL;DR

This paper investigates solutions to Lagrange's equation for large odd integers, demonstrating the existence of solutions where the product of variables plus one has a bounded number of prime factors.

Contribution

It proves that for sufficiently large odd integers, solutions exist with the product of variables plus one having at most 48 prime factors.

Findings

Existence of solutions for large odd N

Bound on prime factors of x_1 x_2 x_3 x_4 + 1

Extension of classical sum of squares results

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.