On squares in polynomial products
Let \(f(X)\in \mathbb{Z}[X]\) be an irreducible polynomial of degree \(D\geq 2\), and let \(n\) be a positive integer. Define \[ F(n)=\prod_{m=1}^{n}f(m). \] For squarefree integer \(d\), nonnegative integers \(M\) and \(N\), we let \(S_d(M,N)\) be the number of integer solutions \((n,s)\) to the equation \[ F(n)=ds^2,\qquad \text{for}\quad n=M+1,\cdots, M+N. \] This paper proved that \[ S_d(M,N)\ll N^{11/12}(\log N)^{1/3} \] uniformly for squarefree integers \(d\geq 1\) and arbitrary integers \(M\geq 0\) and \(N\geq 2\). The above result yields a lower bound on the number of distinct quadratic fields among \(\mathbb{Q}(\sqrt{F(n)})\) for \(n=M+1,\cdots, M+N\).
scientific article; zbMATH DE number 5674289