Perfect powers from products of consecutive terms in arithmetic progression

Abstract

AbstractWe prove that for any positive integersx,dandkwith gcd (x,d)=1 and 3<k<35, the productx(x+d)⋯(x+(k−1)d) cannot be a perfect power. This yields a considerable extension of previous results of Győryet al.and Bennettet al., which covered the cases wherek≤11. We also establish more general theorems for the case wherexcan also be a negative integer and where the product yields an almost perfect power. As in the proofs of the earlier theorems, for fixedkwe reduce the problem to systems of ternary equations. However, our results do not follow as a mere computational sharpening of the approach utilized previously; instead, they require the introduction of fundamentally new ideas. Fork>11, a large number of new ternary equations arise, which we solve by combining the Frey curve and Galois representation approach with local and cyclotomic considerations. Furthermore, the number of systems of equations grows so rapidly withkthat, in contrast with the previous proofs, it is practically impossible to handle the various cases in the usual manner. The main novelty of this paper lies in the development of an algorithm for our proofs, which enables us to use a computer. We apply an efficient, iterated combination of our procedure for solving the new ternary equations that arise with several sieves based on the ternary equations already solved. In this way, we are able to exclude the solvability of the enormous number of systems of equations under consideration. Our general algorithm seems to work for larger values ofkas well, although there is, of course, a computational time constraint.