We discuss the equation in which a, b, and c are non-zero relatively prime integers, p is an odd prime number, and α is a positive integer. The technique used to prove Fermat’s Last Theorem shows that the equation has no solutions with α < 1 or b even. When α=1 and b is odd, there are the two trivial solutions (±1, ∓ 1, ±1). In 1952, Dénes conjectured that these are the only ones. Using methods of Darmon, we prove this conjecture for p≡ 1 mod 4.
@article{bwmeta1.element.bwnjournal-article-aav79i1p7bwm,
author = {Kenneth A. Ribet},
title = {On the equation $a^p + 2^$\alpha$ b^p + c^p = 0$
},
journal = {Acta Arithmetica},
volume = {80},
year = {1997},
pages = {7-16},
zbl = {0877.11015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav79i1p7bwm}
}
Kenneth A. Ribet. On the equation $a^p + 2^α b^p + c^p = 0$
. Acta Arithmetica, Tome 80 (1997) pp. 7-16. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav79i1p7bwm/
[000] [1] B. J. Birch and W. Kuyk (eds.), Modular Functions of One Variable IV, Lecture Notes in Math. 476, Springer, Berlin, 1975.
[001] [2] H. Darmon, The equations and , Internat. Math. Res. Notices 10 (1993), 263-274. | Zbl 0805.11028
[002] [3] H. Darmon, The equation , C. R. Math. Rep. Acad. Sci. Canada 15 (1993), 286-290. | Zbl 0794.11014
[003] [4] H. Darmon, Serre's conjectures, in: Seminar on Fermat's Last Theorem, V. K. Murty (ed.), CMS Conf. Proc. 17, Amer. Math. Soc., Providence, 1995, 135-153.
[004] [5] H. Darmon and A. Granville, On the equations and , Bull. London Math. Soc. 27 (1995), 513-543. | Zbl 0838.11023
[005] [6] P. Dénes, Über die Diophantische Gleichung , Acta Math. 88 (1952), 241-251.
[006] [7] F. Diamond, On deformation rings and Hecke rings, Ann. of Math., to appear. | Zbl 0867.11032
[007] [8] F. Diamond and K. Kramer, Modularity of a family of elliptic curves, Math. Res. Lett. 2 (1995), 299-304. | Zbl 0867.11041
[008] [9] L. E. Dickson, History of the Theory of Numbers, Chelsea, New York, 1971.
[009] [10] L. E. Dickson, Introduction to the Theory of Numbers, University of Chicago Press, Chicago, 1929. | Zbl 55.0092.19
[010] [11] G. Frey, On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2, in: Elliptic Curves, Modular Forms, & Fermat's Last Theorem, J. Coates, S. T. Yau (eds.), International Press, Cambridge, MA, 1995, 79-98. | Zbl 0856.11026
[011] [12] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Grad. Texts in Math. 84, 2nd ed., Springer, Berlin, 1990 | Zbl 0712.11001
[012] [13] S. Kamienny, Rational points on Shimura curves over fields of even degree, Math. Ann. 286 (1990), 731-734. | Zbl 0693.14010
[013] [14] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 33-186. | Zbl 0394.14008
[014] [15] B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162. | Zbl 0386.14009
[015] [16] B. Mazur, Questions about number, in: New Directions in Mathematics, to appear.
[016] [17] F. Momose, Rational points on the modular curves , Compositio Math. 52 (1984), 115-137.
[017] [18] K. A. Ribet, On modular representations of arising from modular forms, Invent. Math. 100 (1990), 431-476. | Zbl 0773.11039
[018] [19] J.-P. Serre, Sur les représentations modulaires de degré 2 de , Duke Math. J. 54 (1987), 179-230.
[019] [20] R. L. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553-572. | Zbl 0823.11030
[020] [21] H. Wasserman, Variations on the exponent-3 Fermat equation, manuscript, 1995.
[021] [22] A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. 141 (1995), 443-551. | Zbl 0823.11029