Almost fifth powers in arithmetic progression
The Diophantine equation \[ x(x+d)\cdots (x+(k-1)d)=y^n \] has been studied by many authors. Recently important results were obtained using the very powerful modular method, but this method needs \(n\geq 7\). Many papers considered the cases \(n=2\) and \(3\), they used elliptic curves and quadratic residues. Here the authors consider the special case \(n=5\) and they use genus 2 curves and apply the Chabauty method. In particular, they prove that the product of \(k\) consecutive terms of a primitive arithmetic progression is never a fifth power when \(3\leq k \leq 54\).
scientific article; zbMATH DE number 5931164
scientific article