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.