Polynomial values of products of terms from an arithmetic progression

L Hajdu,Á Papp

doi:10.1007/s00605-020-01422-7

Abstract

Products of terms of arithmetic progressions yielding a perfect power have been long investigated by many mathematicians. In the particular case of consecutive integers, various finiteness results are known for the polynomial values of such products. In the present paper we consider generalizations of these result in various directions.

Highlights

IntroductionThe first extension of the problem we mention is when on the left hand side of (1), we omit a term from the product, that is, we consider the equation x(x + 1) · · · (x + j − 1)(x + j + 1) · · · (x + k − 1) = yn in positive integers x, k, y, n with k ≥ 2 and n ≥ 2, where 0 ≤ j ≤ k − 1
A classical result of Erdos and Selfridge [11] says that the product of consecutive positive integers is never a perfect power, that is, the equation x(x + 1) · · · (x + k − 1) = yn has no solutions in positive integers x, k, y, n with k ≥ 2 and n ≥ 2
The first extension of the problem we mention is when on the left hand side of (1), we omit a term from the product, that is, we consider the equation x(x + 1) · · · (x + j − 1)(x + j + 1) · · · (x + k − 1) = yn in positive integers x, k, y, n with k ≥ 2 and n ≥ 2, where 0 ≤ j ≤ k − 1

Summary

Introduction

The first extension of the problem we mention is when on the left hand side of (1), we omit a term from the product, that is, we consider the equation x(x + 1) · · · (x + j − 1)(x + j + 1) · · · (x + k − 1) = yn in positive integers x, k, y, n with k ≥ 2 and n ≥ 2, where 0 ≤ j ≤ k − 1. We mention that combining the two directions mentioned above, Saradha and Shorey [22] provided results for equations of the above shape, with one term of the progression missing from the product on the left hand side. To make them work we need to combine several arguments of combinatorial nature, as well

Main results

Proofs