Cubes in products of terms from an arithmetic progression
Volume 184 / 2018
Acta Arithmetica 184 (2018), 117-126
MSC: Primary 11D61.
DOI: 10.4064/aa8655-5-2017
Published online: 14 May 2018
Abstract
We show that there are no cubes in a product with at least $${k-(1-\epsilon)k\frac{\log\log k}{\log k}}, $$ $\epsilon \gt 0,$ terms from a set of $k$ $(\geq 2)$ successive terms in an arithmetic progression having common difference $d$ if either $ k$ is sufficiently large or $3^{\omega(d)}\gg k \frac{\log\log k}{\log k}.$ Here $\omega(d)$ denotes the number of distinct prime divisors of $d.$ This result improves an earlier result of Shorey and Tijdeman.
