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Tbilisi Mathematical Journal

Editor-in-Chief: Inassaridze, Hvedri

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Mathematical Citation Quotient (MCQ) 2016: 0.14

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1512-0139
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The Method of Infinite Ascent Applied on 2p A6 + B3 = C2

Susil Kumar Jena
  • Corresponding author
  • Department of Electronics & Telecommunication Engineering, KIIT University, Bhubaneswar 751024, Odisha, India
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Published Online: 2014-09-29 | DOI: https://doi.org/10.2478/tmj-2014-0003

Abstract

In this paper, we prove that for any positive integer p, when p ≡ 1 (mod 6) or, p ≡ 3 (mod 6), the Diophantine equation: 2p A + B = C has infinitely many co-prime integral solutions A, B, C. When p = 0, this equation has only four integral solutions with (A, B, C) = (±1, 2, ±3). For other integer values of p, the problem is open.

MSC: 11D41; 11D72

Keywords: Diophantine equation 2p A + B = C; method of infinite ascent; higher order Diophantine equation

References

  • [1] S. K. Jena, Method of infinite ascent applied on mA6 + nB3 = C2, Math. Student, Vol. 77 (2008), 239–246.Google Scholar

  • [2] S. K. Jena, Method of infinite ascent applied on A4±nB2 = C3, Math. Student, Vol. 78 (2009), 233–238.Google Scholar

  • [3] S. K. Jena, Method of infinite ascent applied on mA3 + nB3 = C2, Math. Student, Vol. 79 (2010), 187–192.Google Scholar

  • [4] S. K. Jena, Beyond the method of infinite descent, J. Comb. Inf. Syst. Sci., Vol 35 (2010), 501–511.Google Scholar

  • [5] S. K. Jena, Method of infinite ascent applied on mA3 + nB3 = 3C2, Math. Student, Vol. 81 (2012), 151–160.Google Scholar

  • [6] S. K. Jena, The method of infinite ascent applied on A4 ±nB3 = C2, Czech. Math. J. 63 (138) (2013), No. 2, 369-374.Web of ScienceGoogle Scholar

  • [7] S. K. Jena, Method of infinite ascent applied on A3 ± nB2 = C3, Notes on Number Theory and Discrete Mathematics, Vol. 19 (2013), No. 2, 233–238.Google Scholar

  • [8] H. Cohen, Number Theory-Volume I: Tools and Diophantine Equations, Springer-Verlag, GTM 239, 2007.Google Scholar

  • [9] H. Cohen, Number Theory-Volume II: Analytic and Modern Tools, Springer-Verlag, GTM 240, 2007.Google Scholar

  • [10] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, GTM 106, 1986; Expanded 2nd ed., 2009.Google Scholar

About the article

Thankful to my parents - my heavenly mother and revered father, for their subtle support.


Received: 2013-03-24

Accepted: 2014-05-15

Published Online: 2014-09-29


Citation Information: Tbilisi Mathematical Journal, Volume 7, Issue 1, ISSN (Online) 1512-0139, DOI: https://doi.org/10.2478/tmj-2014-0003.

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© 2014 Susil Kumar Jena. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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