Glasnik Matematicki, Vol. 55, No. 1 (2020), 37-53.

PERFECT POWERS IN AN ALTERNATING SUM OF CONSECUTIVE CUBES

Pranabesh Das, Pallab Kanti Dey, Bibekananda Maji and Sudhansu Sekhar Rout

Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada
e-mail: pranabesh.math@gmail.com

Stat-Math Unit, Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi, Delhi - 110016, India
e-mail: pallabkantidey@gmail.com

Department of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, Madhya Pradesh - 453552, India
e-mail: bibekanandamaji@iiti.ac.in

Institute of Mathematics and Applications, Andharua, Bhubaneswar, Odisha - 751029, India
e-mail: lbs.sudhansu@gmail.com

Abstract.   In this paper, we consider the problem about finding out perfect powers in an alternating sum of consecutive cubes. More precisely, we completely solve the Diophantine equation (x+1)3 - (x+2)3 + ∙∙∙ - (x + 2d)3 + (x + 2d + 1)3 = zp, where p is prime and x,d,z are integers with 1 ≤ d ≤ 50.

2010 Mathematics Subject Classification.   11D61, 11D41, 11F11, 11F80

Key words and phrases.   Diophantine equation, Galois representation, Frey curve, modularity, level lowering, linear forms in logarithms

Full text (PDF) (free access)

https://doi.org/10.3336/gm.55.1.04

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