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Cubes in products of terms from an arithmetic progression

Volume 184 / 2018

Pranabesh Das, Shanta Laishram, N. Saradha 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.

Authors

  • Pranabesh DasStat-Math Unit
    Indian Statistical Institute
    Delhi Centre
    New Delhi 110016, India
    e-mail
  • Shanta LaishramStat-Math Unit
    Indian Statistical Institute
    Delhi Centre
    New Delhi 110016, India
    e-mail
  • N. SaradhaINSA Senior Scientist
    Centre for Excellence in Basic Sciences
    Department of Atomic Energy
    University of Mumbai, Kalina Campus
    Mumbai 400098, India
    e-mail

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