A function related to the Mordell–Weil rank of elliptic curves

Let p be a prime number. For each natural number n, we study the behavior of the function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square (mod p). The study of this function is inspired by the cognate function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square. The descent theory of elliptic curves would show that if [Formula: see text] is unbounded for squarefree values of n, then there are elliptic curves over the rational number field with arbitrarily large rank. In this note, we show for every prime p, [Formula: see text] is unbounded as n ranges over squarefree values, thus providing some evidence for the conjecture that [Formula: see text] is unbounded for squarefree n.