Two divisors of (n^2+1)/2 summing up to {\delta}n+{\epsilon}, for {\delta} and {\epsilon} even

TL;DR

This paper investigates the existence of two divisors of (n^2+1)/2 whose sum matches a linear expression with even coefficients, providing complete solutions for specific parameter cases.

Contribution

It fully solves the problem for the cases where δ=2, δ=4, and ε=0, advancing understanding of divisor sums in quadratic forms.

Findings

Solved the divisor sum problem for δ=2 and δ=4 cases.

Provided a complete characterization for ε=0 case.

Enhanced knowledge of divisor properties related to quadratic expressions.

Abstract

In this paper we are dealing with the problem of the existence of two divisors of $(n^2+1)/2$ whose sum is equal to $\delta n+\varepsilon$, in the case when $\delta$ and $\varepsilon$ are even, or more precisely in the case in which $\delta\equiv\varepsilon+2\equiv0$ or $2 \pmod{4}$. We will completely solve the cases $\delta=2, \delta=4$ and $\varepsilon=0$.