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

Siksek, Samir

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.

Résumé
Abstract

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 $p \geq 7$ est premier et $k \geq 2$ . 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 $n \geq 3$ .

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 $p \geq 7$ is prime and $k \geq 2$ . 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 $n \geq 3$ .

MR Zbl

@article{JTNB_2003__15_3_839_0,
     author = {Siksek, Samir},
     title = {On the diophantine equation $x^2 = y^p + 2^k z^p$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {839--846},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {3},
     year = {2003},
     mrnumber = {2142239},
     zbl = {1074.11022},
     language = {en},
     url = {https://www.numdam.org/item/JTNB_2003__15_3_839_0/}
}

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
SP  - 839
EP  - 846
VL  - 15
IS  - 3
PB  - Université Bordeaux I
UR  - https://www.numdam.org/item/JTNB_2003__15_3_839_0/
LA  - en
ID  - JTNB_2003__15_3_839_0
ER  -

%0 Journal Article
%A Siksek, Samir
%T On the diophantine equation $x^2 = y^p + 2^k z^p$
%J Journal de théorie des nombres de Bordeaux
%D 2003
%P 839-846
%V 15
%N 3
%I Université Bordeaux I
%U https://www.numdam.org/item/JTNB_2003__15_3_839_0/
%G en
%F JTNB_2003__15_3_839_0

Siksek, Samir. On the diophantine equation $x^2 = y^p + 2^k z^p$. Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 3, pp. 839-846. https://www.numdam.org/item/JTNB_2003__15_3_839_0/

Bibliographie
Cité par

[1] S.A. Arif, F.S. Abu Muriefah, On the diophantine equation x2 + 2k = yn. Internat. J. Math. & Math. Sci. 20 no. 2 (1997), 299-304. | MR | Zbl

[2] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc. 14 (2001), 843-939. | MR | Zbl

[3] Y. Bugeaud, On the diophantine equation x2 - 2m = ±yn. Proc. Amer. Math. Soc. 125 (1997), 3203-3208. | MR | Zbl

[4] J.E. Cremona, Algorithms for modular elliptic curves (second edition). Cambridge University Press, 1996. | MR | Zbl

[5] J.H.E. Cohn, The diophantine equation x2+2k = yn. Arch. Math. 59 (1992), 341-344. | MR | Zbl

[6] J.H.E. Cohn, The diophantine equation x2+2k = yn, II. Internat. J. Math. & Math. Sci. 22 no. 3 (1999), 459-462. | MR | Zbl

[7] H. Darmon, The equations xn +yn = x2 and xn + yn = z3. International Mathematics Research Notices 10 (1993), 263-274. | MR | Zbl

[8] H. Darmon, L. Merel, Winding quotients and some variants of Format's Last Theorem. J. Reine Angew. Math. 490 (1997), 81-100. | MR | Zbl

[9] F. Diamond, On deformation rings and Hecke rings. Ann. Math. 144 no. 1 (1996), 137-166. | MR | Zbl

[10] Y. Guo, M. Le, A note on the exponential diophantine equation x2 - 2m = yn. Proc. Amer. Math. Soc. 123 (1995), 3627-3629. | MR | Zbl

[11] W. Ivorra, Sur les équations xP + 2βyp = z2 et xP + 2β yp = 2z2. To appear in Acta Arith. | Zbl

[12] A.W. Knapp, Elliptic curves. Mathematical Notes 40, Princeton University Press, 1992. | MR | Zbl

[13] M. Le, On Cohn's conjecture concerning the Diophantine equation x2 + 2m = yn, Arch. Math. 78 no. 1 (2002), 26-35. | MR | Zbl

[14] K. Ribet, On modular representations of Gal(/Q) arising from modular forms. Invent. Math. 100 (1990), 431-476. | MR | Zbl

[15] J.-P. Serre, Sur les répresentations modulaires de degré 2 de Gal(/Q). Duke Math. J. 54 (1987), 179-230. | MR | Zbl

[16] N.P. Smart, The algorithmic resolution of diophantine equations. LMS Student Texts 41, Cambridge University Press, 1998. | MR | Zbl

[17] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. Math. 141 (1995), 553-572. | MR | Zbl

[18] A. Wiles, Modular elliptic curves and Fermat's Last Theorem. Ann. Math. 141 (1995), 443-551. | MR | Zbl