Some Diophantine equations concerning biquadrates

Ajai Choudhry
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 21, 2015, Number 4, Pages 1—5
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Ajai Choudhry
13/4 A Clay Square,
Lucknow, India

Abstract

This paper is concerned with integer solutions of the diophantine equation x14 + x24 + x34 = k x42 where k is a given positive integer. Till now, integer and parametric solutions of this diophantine equation have been published only when k = 1 or 2 or 3. In this paper we obtain parametric solutions of this equation for 43 values of k ≤ 100. We also show that the equation cannot have any solution in integers for 54 values of k ≤ 100. The solvability of the equation x14 + x24 + x34 = k x42 where k could not be determined for three values of k ≤ 100, namely 34, 35 and 65.

Keywords

  • Biquadrates
  • Sums of biquadrates

AMS Classification

  • 11D25

References

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  2. Heath, T. L. (1910) Diophantus of Alexandria, Second Edition, Cambridge University Press.
  3. Piezas, T. A collection of algebraic identities, https://sites.google.com/site/tpiezas/016 (Accessed on 22 May 2015).
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Cite this paper

APA

Choudhry, A. (2015). Some Diophantine equations concerning biquadrates. Notes on Number Theory and Discrete Mathematics, 21(4), 1-5.

Chicago

Choudhry, Ajai. “Some Diophantine Equations concerning Biquadrates.” Notes on Number Theory and Discrete Mathematics 21, no. 4 (2015): 1-5.

MLA

Choudhry, Ajai. “Some Diophantine Equations concerning Biquadrates.” Notes on Number Theory and Discrete Mathematics 21.4 (2015): 1-5. Print.

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