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

top
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

top

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

top
  1. [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. Zbl0881.11038MR1444731
  2. [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. Zbl0982.11033MR1839918
  3. [3] Y. Bugeaud, On the diophantine equation x2 - 2m = ±yn. Proc. Amer. Math. Soc.125 (1997), 3203-3208. Zbl0893.11012MR1422850
  4. [4] J.E. Cremona, Algorithms for modular elliptic curves (second edition). Cambridge University Press, 1996. Zbl0872.14041MR1628193
  5. [5] J.H.E. Cohn, The diophantine equation x2+2k = yn. Arch. Math.59 (1992), 341-344. Zbl0770.11019MR1179459
  6. [6] J.H.E. Cohn, The diophantine equation x2+2k = yn, II. Internat. J. Math. & Math. Sci.22 no. 3 (1999), 459-462. Zbl0960.11025MR1717165
  7. [7] H. Darmon, The equations xn +yn = x2 and xn + yn = z3. International Mathematics Research Notices10 (1993), 263-274. Zbl0805.11028MR1242931
  8. [8] H. Darmon, L. Merel, Winding quotients and some variants of Format's Last Theorem. J. Reine Angew. Math.490 (1997), 81-100. Zbl0976.11017MR1468926
  9. [9] F. Diamond, On deformation rings and Hecke rings. Ann. Math.144 no. 1 (1996), 137-166. Zbl0867.11032MR1405946
  10. [10] Y. Guo, M. Le, A note on the exponential diophantine equation x2 - 2m = yn. Proc. Amer. Math. Soc.123 (1995), 3627-3629. Zbl0852.11016MR1291786
  11. [11] W. Ivorra, Sur les équations xP + 2βyp = z2 et xP + 2β yp = 2z2. To appear in Acta Arith. Zbl1026.11035
  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
  16. [16] N.P. Smart, The algorithmic resolution of diophantine equations. LMS Student Texts41, Cambridge University Press, 1998. Zbl0907.11001MR1689189
  17. [17] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras. Ann. Math.141 (1995), 553-572. Zbl0823.11030MR1333036
  18. [18] A. Wiles, Modular elliptic curves and Fermat's Last Theorem. Ann. Math.141 (1995), 443-551. Zbl0823.11029MR1333035

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.