A note on the quartic Diophantine equation A4 + hB4 = C4 + hD4

Ajai Choudhry
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 23, 2017, Number 1, Pages 1–3
Full paper (PDF, 113 Kb)

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Ajai Choudhry
13/4 A Clay Square
Lucknow 226001, India

Abstract

Integer solutions of the diophantine equation A4 + hB4 = C4 + hD4 are known for all positive integer values of h < 1000. While a solution of the aforementioned diophantine equation for any arbitrary positive integer value of h is not known, Gerardin and Piezas found solutions of this equation when h is given by polynomials of degrees 5 and 2, respectively. In this paper, we present several new solutions of this equation when h is given by polynomials of degrees 2, 3 and 4.

Keywords

  • Biquadrates
  • Fourth powers

AMS Classification

  • 11D25

References

  1. Choudhry, A. (1995) On the Diophantine equation A4 + hB4 = C4 + hD4, Indian J. Pure Appl. Math. 26, 1057–1061.
  2. Dickson, L. E. (1992) History of the Theory of Numbers, Vol. 2, Chelsea Publishing Company, reprint.
  3. Piezas, T. (2013) A collection of algebraic identities, available at https://sites.google.com/site/tpiezas/0021e (accessed on 7 April 2016).
  4. Piezas, T. (2015) On a4 + nb4 = c4 + nd4 and Chebyshev polynomials, available at http://mathoverflow.net/questions/142192/on-a4nb4-c4nd4-and-chebyshev-polynomials (accessed on 7 April 2016).
  5. Tomita, S. http://www.maroon.dti.ne.jp/fermat/dioph121e.html (accessed on 7 April 2016).

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Cite this paper

Choudhry, A. (2017). A note on the quartic Diophantine equation A4 + hB4 = C4 + hD4. Notes on Number Theory and Discrete Mathematics, 23(1), 1-3.

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