A note on the Fermat quartic 34x⁴+y⁴=z⁴

Gustaf Söderlund
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 4, Pages 103–105
DOI: 10.7546/nntdm.2020.26.4.103-105
Full paper (PDF, 156 Kb)

Details

Authors and affiliations

Gustaf Söderlund
Kettilsgatan 4A 58221 Linköping, Sweden

Abstract

We show that the only primitive non-zero integer solutions to the Fermat quartic 34x⁴+y⁴=z⁴ are (x,y,z) = (± 2, ± 3, ± 5). The proofs are based on a previously given complete solution to another Fermat quartic namely x⁴+y⁴=17z⁴.

Keywords

• Fermat quartics
• Diophantine equations
• Primitive non-zero solutions

2010 Mathematics Subject Classification

• 11D41

References

1. Cohen, H. (2007). Number Theory Volume I: Tools and Diophantine Equations, Springer, New York, pp. 397–410 and pp. 462–463.
2. Darmon, H., & Granville, A. (1995). On the equation z^m = F(x, y) and Ax^p + By^q = Cz^r , Bull. London Math. Soc., 27(6), 513–543.
3. Flynn, E. V., & Wetherell, J. L. (2001). Covering collections and a challenge of Serre, Acta Arithmetica, 98, 197–205.
4. Sally J. D., & Sally, P. J. (2007). Roots to Research: A Vertical Development of Mathematical Problems, American Mathematical Society.