Two divisors of $(n^2+1)/2$ summing up to $\delta n+\delta \pm 2$, $\delta$ even

Abstract

We prove there exist infinitely many odd integers $n$ for which there exists a pair of positive divisors $d_1,d_2>1$ of $(n^2+1)/2$ such that $$d_1+d_2=\delta n+(\delta +2).$$ We prove the similar result for $\epsilon =\delta -2$ and $\delta \equiv 4,6(\mathrm{mod}8)$ using different approaches and methods.