On the diophantine equation $x^2=y^p+2^k{z}^p$

Siksek, Samir

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 839-846.

Nous étudions l’équation du titre en utilisant une courbe de Frey, le théorème de descente du niveau de Ribet et une méthode due a Darmon et Merel. Nous pouvons déterminer toutes les solutions entières $x,y,z$, premières deux à deux, si $p\ge 7$ est premier et $k\ge 2$. De cela, nous déduisons des résultats sur quelques cas de cette équation qui ont été étudiés dans la littérature. En particulier, nous pouvons combiner notre résultat avec les résultats précédents de Arif et Abu Muriefah, et avec ceux de Cohn pour obtenir toutes les solutions de l’équation $x^2+2^k=y^n$ pour $n\ge 3$.

We attack the equation of the title using a Frey curve, Ribet’s level-lowering theorem and a method due to Darmon and Merel. We are able to determine all the solutions in pairwise coprime integers $x,y,z$ if $p\ge 7$ is prime and $k\ge 2$. From this we deduce some results about special cases of this equation that have been studied in the literature. In particular, we are able to combine our result with previous results of Arif and Abu Muriefah, and those of Cohn to obtain a complete solution for the equation $x^2+2^k=y^n$ for $n\ge 3$.

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