On the diophantine equation x 2 = y p + 2 k z p

Samir Siksek

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 3, page 839-846
  • ISSN: 1246-7405

Abstract

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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 7 is prime and k 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 3 .

How to cite

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Siksek, Samir. "On the diophantine equation $x^2 = y^p + 2^k z^p$." Journal de théorie des nombres de Bordeaux 15.3 (2003): 839-846. <http://eudml.org/doc/249098>.

@article{Siksek2003,
abstract = {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 $n \ge 3$.},
author = {Siksek, Samir},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {exponential Diophantine equations; elliptic curves},
language = {eng},
number = {3},
pages = {839-846},
publisher = {Université Bordeaux I},
title = {On the diophantine equation $x^2 = y^p + 2^k z^p$},
url = {http://eudml.org/doc/249098},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Siksek, Samir
TI - On the diophantine equation $x^2 = y^p + 2^k z^p$
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 3
SP - 839
EP - 846
AB - 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 $n \ge 3$.
LA - eng
KW - exponential Diophantine equations; elliptic curves
UR - http://eudml.org/doc/249098
ER -

References

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  12. [12] A.W. Knapp, Elliptic curves. Mathematical Notes40, Princeton University Press, 1992. Zbl0804.14013MR1193029
  13. [13] M. Le, On Cohn's conjecture concerning the Diophantine equation x2 + 2m = yn, Arch. Math.78 no. 1 (2002), 26-35. Zbl1006.11013MR1887313
  14. [14] K. Ribet, On modular representations of Gal(/Q) arising from modular forms. Invent. Math.100 (1990), 431-476. Zbl0773.11039MR1047143
  15. [15] J.-P. Serre, Sur les répresentations modulaires de degré 2 de Gal(/Q). Duke Math. J.54 (1987), 179-230. Zbl0641.10026MR885783
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