On the sum of two divisors of (n 2 + 1)/2 Andrej Dujella and Florian Luca 1 Introduction In [2], answering to a question from [1], Ayad and Luca proved that 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. In this paper, we consider the similar problem but with n + 1 replaced by an arbitrary linear polynomial δn + ε, where δ> 0 and ε are given integers. Since the number (n 2 + 1)/2 is odd and both numbers d 1 and d 2 are congruent to 1 modulo 4, it follows that d 1 + d 2 ≡ 2 (mod 4). Hence, if d 1 + d 2 = δn + ε, then either δ ≡ ε ≡ 1 (mod 2), or δ ≡ ε +2 ≡ 0, 2 (mod 4). Here, we will restrict our attention to the first case, namely when both δ and ε are odd. We will give some evidence for the following conjecture. Conjecture 1 If δ> 0 and ε are coprime odd integers and (δ, ε) = (1, 1), then there exist infinitely many positive odd integers n with the property that there exist a pair of positive divisors d 1 and d 2 of (n 2 + 1)/2 with d 1 + d 2 = δn + ε. We prove Conjecture 1 for δ = 1. For general linear polynomials, we give a conditional proof relying on some known conjectures from the distribution of prime numbers. Both our unconditional and conditional proofs rely on known facts from the theory of Pell equations. 2 Monic polynomials – parametric solution In this section, we look at polynomials of the form n + ε, where ε is an odd integer. We will show that the polynomial n + 1 studied in [2] is the unique polynomial of this form for which there do not exist n, d 1 and d 2 with the property that we are considering. 1