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

Abstract

Integer solutions of the diophantine equation A4hB4C4hD4 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 A4hB4C4hD4, 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

APA

Choudhry, A. (2017). A note on the quartic Diophantine equation A4hB4C4hD4. Notes on Number Theory and Discrete Mathematics, 23(1), 1-3.

Chicago

Choudhry, Ajai. “A Note on the Quartic Diophantine Equation A4hB4C4hD4.” Notes on Number Theory and Discrete Mathematics 23, no. 1 (2017): 1-3.

MLA

Choudhry, Ajai. “A Note on the Quartic Diophantine Equation A4hB4C4hD4.” Notes on Number Theory and Discrete Mathematics 23.1 (2017): 1-3. Print.

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