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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## Details

### Authors and affiliations

Ajai Choudhry

*13/4 A Clay Square
Lucknow 226001, India
*

### Abstract

Integer solutions of the diophantine equation *A*^{4} + *hB*^{4} = *C*^{4} + *hD*^{4} 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

- Choudhry, A. (1995) On the Diophantine equation
*A*^{4}+*hB*^{4}=*C*^{4}+*hD*^{4}, Indian J. Pure Appl. Math. 26, 1057–1061. - Dickson, L. E. (1992) History of the Theory of Numbers, Vol. 2, Chelsea Publishing Company, reprint.
- Piezas, T. (2013) A collection of algebraic identities, available at https://sites.google.com/site/tpiezas/0021e (accessed on 7 April 2016).
- Piezas, T. (2015) On
*a*^{4}+*nb*^{4}=*c*^{4}+*nd*^{4}and Chebyshev polynomials, available at http://mathoverflow.net/questions/142192/on-a4nb4-c4nd4-and-chebyshev-polynomials (accessed on 7 April 2016). - Tomita, S. http://www.maroon.dti.ne.jp/fermat/dioph121e.html (accessed on 7 April 2016).

## Related papers

## Cite this paper

APAChoudhry, A. (2017). A note on the quartic Diophantine equation *A*^{4} + *hB*^{4} = *C*^{4} + *hD*^{4}. Notes on Number Theory and Discrete Mathematics, 23(1), 1-3.

Choudhry, Ajai. “A Note on the Quartic Diophantine Equation *A*^{4} + *hB*^{4} = *C*^{4} + *hD*^{4}.” Notes on Number Theory and Discrete Mathematics 23, no. 1 (2017): 1-3.

Choudhry, Ajai. “A Note on the Quartic Diophantine Equation *A*^{4} + *hB*^{4} = *C*^{4} + *hD*^{4}.” Notes on Number Theory and Discrete Mathematics 23.1 (2017): 1-3. Print.