\(p\)-adic valuation of \(\prod_{k=m+1}^n (k^2-m^2)\) (Q6546721)

The authors present a formula for the exponent of the prime \(p\) in \N\[\NC_{k,m}=\prod_{k=m+1}^n (k^2-m^2). \N\]\NTheir formula can be deduced from the classical formula for the exponent of a prime in the factorization of a factorial by noticing that \(C_{k,m}\) is a product of two rational numbers each of which is a ratio of two factorials. They also show that \(C_{2n}\) and \(C_{3,n}\) are not squares for \(n\ge 3\) and \(n\ge 4\), respectively.