Squares in (2^2-1)...(n^2-1) and p-adic valuation

TL;DR

This paper characterizes all perfect squares in a specific sequence derived from products of quadratic expressions and provides a formula for their p-adic valuations, revealing infinitely many squares in the sequence.

Contribution

It explicitly determines all squares in the sequence and introduces a formula for their p-adic valuations, advancing understanding of the sequence's properties.

Findings

All squares in the sequence are explicitly characterized.

There are infinitely many squares in the sequence.

A formula for the p-adic valuation of sequence terms is provided.

Abstract

In this paper, we determine all the squares in the sequence $\{\prod_{k=2}^n(k^2-1)\}_{n=2}^\infty $. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the $p$-adic valuation of the terms in this sequence.