Theoretical Statistics and Mathematics Unit, ISI Delhi

Cubes in products of terms from an arithmetic progression

by Pranabesh Das, Shanta Laishram and N. Saradha

We show that there are no cubes in a product with at least
$k-(1-\epsilon)k\frac{\log\log k}{\log k},\epsilon>0,$ 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.

isid/ms/2016/16 [fulltext]

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