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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## Details

### 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

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

## Related papers

## Cite this paper

APAIzadi, F. & A. S. Zargar. (2014). On integer solutions of *A*^{5} + *B*^{3} = *C*^{5} + *D*^{3} Notes on Number Theory and Discrete Mathematics, 20(5), 20-24.

Izadi, Farzali, and Arman Shamsi Zargar. “On Integer Solutions of *A*^{5} + *B*^{3} = *C*^{5} + *D*^{3}.” Notes on Number Theory and Discrete Mathematics 20, no. 5 (2014): 20-24.

Izadi, Farzali, and Arman Shamsi Zargar. “On Integer Solutions of *A*^{5} + *B*^{3} = *C*^{5} + *D*^{3}.” Notes on Number Theory and Discrete Mathematics 20.5 (2014): 20-24. Print.