On the equation $a^p + 2^α b^p + c^p = 0$

Kenneth Ribet

ArticleOriginal scientific text

Title

On the equation $a^{p} + 2^{α} b^{p} + c^{p} = 0$

Authors ¹

Affiliations

Department of Mathematics, University of California, Berkeley, California 94720-3840, U.S.A.

Abstract

We discuss the equation

a^{p} + 2^{α} b^{p} + c^{p} = 0

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.

B. J. Birch and W. Kuyk (eds.), Modular Functions of One Variable IV, Lecture Notes in Math. 476, Springer, Berlin, 1975.
H. Darmon, The equations $x^{n} + y^{n} = z ²$ and $x^{n} + y^{n} = z ³$ , Internat. Math. Res. Notices 10 (1993), 263-274.
H. Darmon, The equation $x ⁴ - y ⁴ = z^{p}$ , C. R. Math. Rep. Acad. Sci. Canada 15 (1993), 286-290.
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.
H. Darmon and A. Granville, On the equations $z^{m} = F (x, y)$ and $A x^{p} + B y^{q} = C z^{r}$ , Bull. London Math. Soc. 27 (1995), 513-543.
P. Dénes, Über die Diophantische Gleichung $x^{l} + y^{l} = c z^{l}$ , Acta Math. 88 (1952), 241-251.
F. Diamond, On deformation rings and Hecke rings, Ann. of Math., to appear.
F. Diamond and K. Kramer, Modularity of a family of elliptic curves, Math. Res. Lett. 2 (1995), 299-304.
L. E. Dickson, History of the Theory of Numbers, Chelsea, New York, 1971.
L. E. Dickson, Introduction to the Theory of Numbers, University of Chicago Press, Chicago, 1929.
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.
K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Grad. Texts in Math. 84, 2nd ed., Springer, Berlin, 1990
S. Kamienny, Rational points on Shimura curves over fields of even degree, Math. Ann. 286 (1990), 731-734.
B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 33-186.
B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.
B. Mazur, Questions about number, in: New Directions in Mathematics, to appear.
F. Momose, Rational points on the modular curves $X_{s p l i t} (p)$ , Compositio Math. 52 (1984), 115-137.
K. A. Ribet, On modular representations of $G a l (ℚ \frac{̅}{ℚ})$ arising from modular forms, Invent. Math. 100 (1990), 431-476.
J.-P. Serre, Sur les représentations modulaires de degré 2 de $G a l (ℚ \frac{̅}{ℚ})$ , Duke Math. J. 54 (1987), 179-230.
R. L. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553-572.
H. Wasserman, Variations on the exponent-3 Fermat equation, manuscript, 1995.
A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. 141 (1995), 443-551.

Title

On the equation αap+2αbp+cp=0

Affiliations

Abstract

Bibliography

On the equation $a^{p} + 2^{α} b^{p} + c^{p} = 0$