Glasnik matematički, Vol. 55 No. 1, 2020.
Original scientific paper
https://doi.org/10.3336/gm.55.1.04
Perfect powers in an alternating sum of consecutive cubes
Pranabesh Das
orcid.org/0000-0001-9119-5402
; Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada
Pallab Kanti Dey
; Stat-Math Unit, Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi, Delhi - 110016, India
Bibekananda Maji
orcid.org/0000-0003-2155-2480
; Department of Mathematics, Indian Institute of Technology Indore, Simrol, Indore, Madhya Pradesh - 453552, India
Sudhansu Sekhar Rout
; Institute of Mathematics and Applications, Andharua, Bhubaneswar, Odisha - 751029, India
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.
Keywords
Diophantine equation; Galois representation; Frey curve; modularity; level lowering; linear forms in logarithms
Hrčak ID:
239041
URI
Publication date:
12.6.2020.
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