## The Fermat-type equations $x^5 + y^5 = 2z^p$ or $3z^p$ solved through $\mathbb {Q}$-curves

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- by Luis Dieulefait and Nuno Freitas PDF
- Math. Comp.
**83**(2014), 917-933 Request permission

## Abstract:

We solve the Diophantine equations $x^5 + y^5 = dz^p$ with $d=2, 3$ for a set of prime numbers of density $3/4$. The method consists of relating a possible solution to another Diophantine equation and solving the latter via a generalized modular technique. Indeed, we will apply a multi-Frey technique with two $\mathbb {Q}$-curves along with a new technique for eliminating newforms.## References

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## Additional Information

**Luis Dieulefait**- Affiliation: Department of Algebra and Geometry, University of Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
- MR Author ID: 671876
**Nuno Freitas**- Affiliation: Department of Algebra and Geometry, University of Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
- MR Author ID: 1044711
- Received by editor(s): May 24, 2011
- Received by editor(s) in revised form: November 26, 2011, December 15, 2011, January 13, 2012, March 6, 2012, and June 2, 2012
- Published electronically: June 10, 2013
- Additional Notes: The first author’s research was supported by project MICINN MTM2009-07024 from MECD, Spain; and ICREA Academia Research Prize.

The second author’s research was supported by a scholarship from Fundaçao para a Ciência e a Tecnologia, Portugal, reference no. $SFRH/BD/44283/2008$. - © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**83**(2014), 917-933 - MSC (2010): Primary 11D41
- DOI: https://doi.org/10.1090/S0025-5718-2013-02731-7
- MathSciNet review: 3143698