Diophantine equations with products of consecutive values of a quadratic polynomial
Let \(a,b,c,d\) be fixed nonnegative integers with \(a,d \geq 1\) and \(\gcd (a,b,c)=1\). Suppose that \(f(X)=aX^2+bX+c\) is an irreducible polynomial over \(\mathbb Q\). The authors prove that the solutions of the Diophantine equation \[ \prod_{k=1} ^n f(k)=dy^l, \quad l,n,y \in {\mathbb N}, \;l\geq 2, \] verify \[ n\leq \max \{ 10^8, b,c,\frac{d}{2a}, n_0\}, \] where \(n_0\) is explicitly determined in terms of \(a\), \(b\), \(c\). One of the consequences of this result is that the only positive integer solution to the equation \(\prod_{k=1} ^n (2k^2+7)=5y^l\) is \((n,y,l)=(2,3,3)\). The proof uses the idea of \textit{J. Cilleruelo} [J. Number Theory 128, No. 8, 2488--2491 (2008; Zbl 1213.11057)] and various inequalities concerning the distribution of prime numbers.
arithmetic progressions of prime numbers
scientific article; zbMATH DE number 5931158
scientific article