Almost fifth powers in arithmetic progression

We prove that the product of k consecutive terms of a primitive arithmetic progression is never a perfect fifth power when 3⩽k⩽54. We also provide a more precise statement, concerning the case where the product is an “almost” fifth power. Our theorems yield considerable improvements and extensions, in the fifth power case, of recent results due to Győry, Hajdu and Pintér. While the earlier results have been proved by classical (mainly algebraic number theoretical) methods, our proofs are based upon a new tool: we apply genus 2 curves and the Chabauty method (both the classical and the elliptic verison).