Powers from Products of Consecutive Terms in Arithmetic Progression

Abstract

We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

Powers from products of $k$ terms in progression: finiteness for small $k$

“…For small values of k, finiteness results for (1.1) (under coprimality assumptions) have been obtained for k ≤ 82 in [1,Theorem 1.4] and, in completely explicit form for k ≤ 34, in [10]. The techniques of [1] do not allow for substantial strengthening of these results, however.…”

“…The techniques of [1] do not allow for substantial strengthening of these results, however. The goal of the paper at hand is to considerably extend [1, Theorem 1.4] using techniques from [2] and a variety of new ideas.…”

The Least Common Multiple of Consecutive Terms in a Quadratic Progression

“…See [2,15] for some other results. However, there are few renowned theorems but more conjectures about quadratic progressions, among which the sequence {n 2 + 1} n∈N is best known.…”

On a generalization of a problem of Erdos and Graham

“…Laishram and Shorey [23] extended this result to the case where either d ≤ 10 10 , or d has at most six prime divisors. Bennett, Bruin, Győry and Hajdu [3] solved (1) with 6 ≤ k ≤ 11 and l = 2. Hirata-Kohno, Laishram, Shorey and Tijdeman [22] completely solved (1) with 3 ≤ k < 110.…”