engleski

Two divisors of (n^2+1)/2 summing up to delta*n + delta ± 2, delta even

Ayad and Luca have proved that there does not exist an odd integer n>1 and two positive divisors d_1, d_2 of (n^2+1)/2 such that d_1+d_2=n+1. Dujella and Luca have dealt with more general issue, where n+1 was replaced with an arbitrary linear polynomial delta*n+epsilon, when delta>0 and epsilon are given integers and they have focused on the case when delta and epsilon are odd. We deal with one-parametric families of even coefficients delta and epsilon, when epsilon = delta + 2 and epsilon = delta - 2 and we prove that there exist infinitely many odd integers n with the property that there exists a pair of positive divisors d_1, d_2 of (n^2+1)/2 such that d_1+d_2=delta*n+(delta+2). We also prove that there exist infinitely many odd integers n with the property that there exists a pair of positive divisors d_1, d_2 of (n^2+1)/2 such that d_1+d_2=delta*n+(delta-2), delta==4, 6 (mod 8). We use Diophantine equations and their properties, with a special accent on Pellian equations and their properties.

sum of divisors; Legendre symbol; quadratic congruences

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