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

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 $\varepsilon=0$.