Geodesic
Perfect powers expressible as sums of two fifth or seventh powers
Sander R. Dahmen 1   ; Samir Siksek 2
1 Department of Mathematics VU University Amsterdam De Boelelaan 1081a 1081 HV Amsterdam, The Netherlands
2 Mathematics Institute University of Warwick Coventry, CV4 7AL, United Kingdom
Acta Arithmetica, Tome 164 (2014) no. 1, pp. 65-100 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We show that the generalized Fermat equations with signatures $(5,5,7)$, $(5,5,19)$, and $(7,7,5)$ (and unit coefficients) have no non-trivial primitive integer solutions. Assuming GRH, we also prove the non-existence of non-trivial primitive integer solutions for the signatures $(5,5,11)$, $(5,5,13)$, and $(7,7,11)$. The main ingredients for obtaining our results are descent techniques, the method of Chabauty–Coleman, and the modular approach to Diophantine equations.
DOI : 10.4064/aa164-1-5
Keywords: generalized fermat equations signatures unit coefficients have non trivial primitive integer solutions assuming grh prove non existence non trivial primitive integer solutions signatures main ingredients obtaining results descent techniques method chabauty coleman modular approach diophantine equations

Sander R. Dahmen  1   ; Samir Siksek  2

1 Department of Mathematics VU University Amsterdam De Boelelaan 1081a 1081 HV Amsterdam, The Netherlands
2 Mathematics Institute University of Warwick Coventry, CV4 7AL, United Kingdom 
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Sander R. Dahmen; Samir Siksek. Perfect powers expressible as sums of
 two fifth or seventh powers. Acta Arithmetica, Tome 164 (2014) no. 1, pp. 65-100. doi: 10.4064/aa164-1-5

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