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) (access from subscribing institutions only)

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

References:

  1. M. Bennett, K. Györy and Á. Pintér, On the diophantine equation 1 k + 2 k + ∙∙∙ + x k = yn, Compos. Math. 140 (2004), 1417-1431.
    MathSciNet     CrossRef

  2. M. Bennett, A superelliptic equation involving alternating sums of powers, Publ. Math. Debrecen 79 (2011), 317-324.
    MathSciNet     CrossRef

  3. M. Bennett, V. Patel and S. Siksek, Perfect powers that are sums of consecutive cubes, Mathematika 63 (2017), 230-249.
    MathSciNet     CrossRef

  4. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symboli Comput. 24 (1997), 235-265.
    MathSciNet     CrossRef

  5. J. W. S. Cassels, A Diophantine equation, Glasgow Math. J. 27 (1985), 11-88.
    MathSciNet     CrossRef

  6. H. Cohen, Number theory vol. II: analytic and modern tools, Springer, New York, 2007.
    MathSciNet    

  7. K. Dilcher, On a Diophantine equation involving quadratic characters, Compos. Math. 57 (1986), 383-403.
    MathSciNet     CrossRef

  8. M. Jacobson, Á. Pintér, P. G. Walsh, A computational approach for solving y2= 1 k + 2 k + ∙∙∙ + x k, Math Comp. 72 (2003), 2099-2110.
    MathSciNet     CrossRef

  9. A. Kraus, Majorations effectives pour l'équation de Fermat généralisée, Canad. J. Math. 49 (1997), 1139-1161.
    MathSciNet     CrossRef

  10. M. Laurent, Linear forms in two logarithms and interpolation determinants II, Acta. Arith. 133 (2008), 325-348.
    MathSciNet     CrossRef

  11. W. J. Leveque, On the equation ym = f(x), Acta. Arith. 9 (1964), 209-219.
    MathSciNet     CrossRef

  12. É. Lucas, Problem 1180, Nouvelle Ann. Math. 14 (1875), 336.

  13. B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.
    MathSciNet     CrossRef

  14. Á. Pintér, On the power values of power sums, J. Number Theory 125 (2007), 412-423.
    MathSciNet     CrossRef

  15. K. Ribet, On modular representations of Gal( ̅ℚ/ℚ) arising from modular forms, Invent. Math. 100 (1990), 431-476.
    MathSciNet     CrossRef

  16. J. J. Schäffer, The equation 1 p + 2 p + ∙∙∙+ n p = m q, Acta Math. 95 (1956), 155-189.
    MathSciNet     CrossRef

  17. A. Schinzel and R. Tijdeman, On the equation ym = f(x), Acta Arith. 31 (1976), 199-204.
    MathSciNet     CrossRef

  18. S. Siksek, Modular approach to Diophantine equations, in: Explicit methods in number theory, Panor. Syntheses 36, Soc. Math. France, Paris, 2012, 151–-179.
    MathSciNet    

  19. N. P. Smart, The algorithmic resolution of Diophantine equations, Cambridge University Press, 1997.
    MathSciNet     CrossRef

  20. G. Soydan, On the Diophantine equation (x+1)k + (x+2)k + ∙∙∙ + (lx)k = yn, Publ. Math. Debrecen 91 (2017), 369-382.
    MathSciNet     CrossRef

  21. R. J. Stroeker, On the sum of consecutive cubes being a square, Compos. Math. 97 (1995), 295-307.
    MathSciNet     CrossRef

  22. G. N. Watson, The problem of the square pyramid, Messenger of Math. 48 (1918), 1-22.

  23. A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. Math. (2) 141 (1995), 443-551.
    MathSciNet     CrossRef

  24. Z. Zhang and M. Bai, On the diophantine equation (x+1) 2 + (x+2) 2 + ∙∙∙+ (x+d) 2 = yn, Funct. Approx. Comment. Math. 49 (2013), 73-77.
    MathSciNet     CrossRef

  25. Z. Zhang, On the Diophantine equation (x-1)k + xk + (x+1)k = yn, Publ. Math. Debrecen 85 (2014), 93-100.
    MathSciNet     CrossRef
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