On integer solutions of A5 + B3 = C5 + D3

Farzali Izadi and Arman Shamsi Zargar
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 20, 2014, Number 5, Pages 20—24
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Authors and affiliations

Farzali Izadi
Department of Pure Mathematics, Azarbaijan Shahid Madani University
Tabriz 53751-71379, Iran

Arman Shamsi Zargar
Department of Pure Mathematics, Azarbaijan Shahid Madani University
Tabriz 53751-71379, Iran

Abstract

In this note, we study the diagonal nonhomogeneous symmetric Diophantine equation of the title, and show that when a solution has been found, a series of other solutions can be derived. This shows that difference of quintics equals difference of cubics for infinitely many integers. We do so using a method involving elliptic curves, which makes it possible to naturally find any solution in a matter of minutes.

Keywords

  • Diophantine equation
  • Elliptic curve

AMS Classification

  • 11D25
  • 11G05

References

  1. Bremner, A., Bremner, A., M. Ulas, On xa ± yb ± zc ± wd =0, 1/a+1/b+1/c+1/d=1, Int. J. Number Theory, Vol. 7, 2011, 2081–2090
  2. Choudhry, A, Symmetric Diophantine equations, Rocky Mountain J. Math. , Vol. 34, 2004, 1281–1298.
  3. Dickson, L. E., History of the Theory of Numbers II, Chelsea Publishing Company, New York, 1920.
  4. Sage software, Version 4.5.3, http://www.sagemath.org
  5. Washington, L. C., Elliptic Curves: Number Theory and Cryptography, 2nd ed., CRC Press, Taylor & Francis Group, Boca Raton, FL, 2008.

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Cite this paper

APA

Izadi, F. & A. S. Zargar. (2014). On integer solutions of A5 + B3 = C5 + D3 Notes on Number Theory and Discrete Mathematics, 20(5), 20-24.

Chicago

Izadi, Farzali, and Arman Shamsi Zargar. “On Integer Solutions of A5 + B3 = C5 + D3.” Notes on Number Theory and Discrete Mathematics 20, no. 5 (2014): 20-24.

MLA

Izadi, Farzali, and Arman Shamsi Zargar. “On Integer Solutions of A5 + B3 = C5 + D3.” Notes on Number Theory and Discrete Mathematics 20.5 (2014): 20-24. Print.

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