Gustav Söderlund

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 23, 2017, Number 2, Pages 36—44

**Download full paper: PDF, 181 Kb**

## Details

### Authors and affiliations

Gustav Söderlund

*Kettilsgatan 4A 582 21 Linköping, Sweden
*

### Abstract

We find all primitive non-zero integer solutions to the title equation, namely (*x*, *y*, *z*) =(±5,±3, 11). The proofs involved are based solely on elementary methods with no use of computers and the elliptic curve machinery.

### Keywords

- Diophantine equations
- Primitive non-zero solutions

### AMS Classification

- 11D41

### References

- Terai, N. & Osada, H. (1992) The Diophantine equation
*x*^{4}+*d**y*^{4}=*z*^{p}, C.R. Math. Rep. Acad. Sci. Canada, 14(1), 55–58. - Cao, Z. F. (1992) The Diophantine equation
*c**x*^{4}+*d**y*^{4}=*z*^{p}C.R. Math. Rep. Acad. Sci.

Canada, 14(5), 231–234. - Darmon, H. & Granville, A. (1995) On the equation
*z*^{m}= F(*x, y*) and*A x*^{p}+*B y*^{q}=*C z*^{r}, Bull. London Math. Soc., 27(6), 513–543. - Bennett, M. A., Ellenberg, J., S., & Ng, N. (2010) The Diophantine equation
*A*^{4}+ 2^{ β }*B*^{ 2 }=*C*^{ p }, Inter. J. Number Theory, 6, 1–27. - Dieulefait, L., & Urroz, J. J. (2009) Solving Fermat-type equations via modular Q-curves over polyquadratic fields, J. Crelle, 633, 183–195.
- Bruin, N. (1999) The Diophantine equation
*x*^{2}±*y*^{4}= ±*z*^{6}and*x*^{2}+*y*^{8}=*z*^{3}, Compositio Math., 118, 305–321. - Bruin, N. (2005) The primitive solutions to
*x*^{3}+*y*^{9}=*z*^{2}, J. Numb. Theor., 111, 179–189. - Poonen, B., Schaefer, E., & Stoll, M. (2007) Twists of
*X*(7) and primitive solutions to*x*^{2}+*y*^{7}=*z*^{3}, Duke Math. J., 137(1), 103–158. - Söderlund, G., & Aldén, E. (2013) A note on the diophantine equation
*C z*^{2}=*x*^{5}+*y*^{5}, Advances in Theoretical and Applied Mathematics, 8(3), 173–176. - Mauldin, R., D. (1997) A generalization of Fermat’s last theorem: the Beale conjecture and prize problem, Notices Amer. Math. Soc., 44(11), 1436–1437.
- Goldfeld, D. (2002) Modular forms, elliptic curves and the abc conjecture, in: A panorama of number theory or the view from Baker’s garden, Wüstholz, G., (ed.), 173–189.
- Lucas, E. (1877) Sur la résolution du système des équations 2
*v*^{2}−*u*^{2}=*w*^{2}et 2*v*^{2}+*u*^{2}= 3*z*^{2}en nombre entiers, Nouvelles annales de mathématiques 2e série, tome 16, 409–416. http://www.numdam.org/numdam-bin/browse?j=NAM\&sl=0 - Sally, J., D., & Sally, P., J. (2007) Roots to Research: A vertical Development of Mathematical problems, American Mathematical Society.
- Cohen, H. (2007) Number Theory Volume I: Tools and Diophantine equations, Springer, New York.

## Related papers

## Cite this paper

Söderlund, G. (2017). The primitive solutions to the Diophantine equation 2*X*^{4} + *Y*^{4} =*Z*^{3}. Notes on Number Theory and Discrete Mathematics, 23(2), 36-44.