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

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: bazsoa@science.unideb.hu

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 bk +(a+b)k + ··· + (a(x-1) + b)k= dl + (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.


Full text (PDF) (free access)

DOI: 10.3336/gm.47.2.02


References:

  1. A. Bazsó, Á. Pintér and H. M. Srivastava, A refinement of Faulhaber's Theorem con\-cern\-ing sums of powers of natural numbers, Appl. Math. Lett. 25 (2012), 486-489.
    MathSciNet     CrossRef

  2. Y. F. Bilu and R. F. Tichy, The Diophantine equation f(x) = g(y), Acta Arith. 95 (2000), 261-288.
    MathSciNet    

  3. Y. F. Bilu, B. Brindza, P. Kirschenhofer, \'A. Pintér and R. F. Tichy, Diophantine equations and Bernoulli polynomials (with an Appendix by A. Schinzel), Compositio Math. 131 (2002), 173-188.
    MathSciNet     CrossRef

  4. J. Brillhart, On the Euler and Bernoulli polynomials, J. Reine Angew. Math. 234 (1969), 45-64.
    MathSciNet     CrossRef

  5. B. Brindza, On S-integral solutions of the equation ym=f(x), Acta Math. Hungar. 44 (1984), 133-139.
    MathSciNet     CrossRef

  6. Á. Pintér and Cs. Rakaczki, On the zeros of shifted Bernoulli polynomials, Appl. Math. Comput. 187 (2007), 379-383.
    MathSciNet     CrossRef

  7. H. Rademacher, Topics in Analityc Number Theory, Springer-Verlag, Berlin, 1973.
    MathSciNet     CrossRef

  8. Cs. Rakaczki, On some generalizations of the Diophantine equation s(1k+2k+ ··· +xk)+r=dyn Acta Arith. 151 (2012), 201-216.
    MathSciNet     CrossRef

  9. J. J. Schäffer, The equation 1p+2p+3p+ ··· +np=mq, Acta Math. 95 (1956), 155-189.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page