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Added link to my paper.
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Michael Stoll
Michael Stoll
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To the best of my knowledge, this is open for general $p$. As mentioned by Alapan Das, Bjorn Poonen has solved the case $p = 2$ and also $p = 3$ [B. Poonen, Some diophantine equations of the form $x^n + y^n = z^m$, Acta Arith. 86 (1998), 193-205]. The case $p = 5$ is part of FLT. Sander Dahmen and Samir Siksek [Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164 (2014), 65-100] solve the cases $p = 7$ and $p = 19$, and also, assuming GRH, $p = 11, 13$. In my paper "Chabauty without the Mordell-Weil group""Chabauty without the Mordell-Weil group" [In: G. Böckle, W. Decker, G: Malle (Eds.): Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, Springer Verlag (2018)] I remove the GRH assumption on these two cases and do also $p = 17$, and I extend the range of primes for which the equation can be solved under GRH to $p \le 53$. In all these cases, no nontrivial solutions exist.

To the best of my knowledge, this is open for general $p$. As mentioned by Alapan Das, Bjorn Poonen has solved the case $p = 2$ and also $p = 3$ [B. Poonen, Some diophantine equations of the form $x^n + y^n = z^m$, Acta Arith. 86 (1998), 193-205]. The case $p = 5$ is part of FLT. Sander Dahmen and Samir Siksek [Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164 (2014), 65-100] solve the cases $p = 7$ and $p = 19$, and also, assuming GRH, $p = 11, 13$. In my paper "Chabauty without the Mordell-Weil group" [In: G. Böckle, W. Decker, G: Malle (Eds.): Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, Springer Verlag (2018)] I remove the GRH assumption on these two cases and do also $p = 17$, and I extend the range of primes for which the equation can be solved under GRH to $p \le 53$. In all these cases, no nontrivial solutions exist.

To the best of my knowledge, this is open for general $p$. As mentioned by Alapan Das, Bjorn Poonen has solved the case $p = 2$ and also $p = 3$ [B. Poonen, Some diophantine equations of the form $x^n + y^n = z^m$, Acta Arith. 86 (1998), 193-205]. The case $p = 5$ is part of FLT. Sander Dahmen and Samir Siksek [Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164 (2014), 65-100] solve the cases $p = 7$ and $p = 19$, and also, assuming GRH, $p = 11, 13$. In my paper "Chabauty without the Mordell-Weil group" [In: G. Böckle, W. Decker, G: Malle (Eds.): Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, Springer Verlag (2018)] I remove the GRH assumption on these two cases and do also $p = 17$, and I extend the range of primes for which the equation can be solved under GRH to $p \le 53$. In all these cases, no nontrivial solutions exist.

Source Link
Michael Stoll
Michael Stoll
  • 11.3k
  • 2
  • 45
  • 60

To the best of my knowledge, this is open for general $p$. As mentioned by Alapan Das, Bjorn Poonen has solved the case $p = 2$ and also $p = 3$ [B. Poonen, Some diophantine equations of the form $x^n + y^n = z^m$, Acta Arith. 86 (1998), 193-205]. The case $p = 5$ is part of FLT. Sander Dahmen and Samir Siksek [Perfect powers expressible as sums of two fifth or seventh powers, Acta Arith. 164 (2014), 65-100] solve the cases $p = 7$ and $p = 19$, and also, assuming GRH, $p = 11, 13$. In my paper "Chabauty without the Mordell-Weil group" [In: G. Böckle, W. Decker, G: Malle (Eds.): Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, Springer Verlag (2018)] I remove the GRH assumption on these two cases and do also $p = 17$, and I extend the range of primes for which the equation can be solved under GRH to $p \le 53$. In all these cases, no nontrivial solutions exist.