Glasnik Matematicki, Vol. 47, No. 2 (2012), 253-263.

ON EQUAL VALUES OF POWER SUMS OF ARITHMETIC PROGRESSIONS

András Bazsó, Dijana Kreso, Florian Luca and Ákos Pintér

Institut für Mathematik (A), Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria
e-mail: kreso@math.tugraz.at

Mathematical Center UNAM, UNAM Ap. Postal 61-3 (Xangari), CP 58 089, Morelia, Michoacán, Mexico
e-mail: fluca@matmor.unam.mx

Institute of Mathematics, MTA-DE Research Group "Equations, functions and curves", Hungarian Academy of Science , University of Debrecen, H-4010 Debrecen, P.O. Box 12, Hungary
e-mail: apinter@science.unideb.hu

Abstract.   In this paper, we consider the Diophantine equation b^k +(a+b)^k + ··· + (a(x-1) + b)^k= d^l + (c+d)^l + ··· + (c(y-1) + d)^l, where a,b,c,d,k,l are given integers with gcd (a,b) = gcd (c,d) = 1, k ¹ l. We prove that, under some reasonable assumptions, the above equation has only finitely many solutions.

2010 Mathematics Subject Classification.   11B68, 11D41.

Key words and phrases.   Diophantine equations, exponential equations, Bernoulli polynomials.

DOI: 10.3336/gm.47.2.02

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