Two divisors of \((n^2+1)/2\) summing up to \(n+1\)
In [Complex Var. Elliptic Equ. 51, No. 2, 143--160 (2006; Zbl 1091.12001)], \textit{M. Ayad} asked to prove that there does not exist an odd prime \(p\) and two positive divisors \(d_1\) and \(d_2\) of \((p^2+1)=2\) such that \(d_1+d_2 = p+1\). In this note, the authors prove the following: Theorem. There does not exist an odd integer \(n > 1\) and two positive divisors \(d_1\) and \(d_2\) of \((n^2 + 1)=2\) such that \(d_1 + d_2 = n+1\). Two applications for this result are given, the first deals with the indecomposability of a specific polynomial (already proved in the paper cited above). The second shows that a certain Diophantine equation can have infinitely many integer solutions.
scientific article; zbMATH DE number 5302614