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## Acta Arithmetica

1997 | 79 | 1 | 7-16
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
7-16
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-08-11
Twórcy
autor
• Department of Mathematics, University of California, Berkeley, California 94720-3840, U.S.A.
Bibliografia
• [1] B. J. Birch and W. Kuyk (eds.), Modular Functions of One Variable IV, Lecture Notes in Math. 476, Springer, Berlin, 1975.
• [2] H. Darmon, The equations $x^n + y^n = z²$ and $x^n + y^n = z³$, Internat. Math. Res. Notices 10 (1993), 263-274.
• [3] H. Darmon, The equation $x⁴-y⁴ = z^p$, C. R. Math. Rep. Acad. Sci. Canada 15 (1993), 286-290.
• [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.
• [5] H. Darmon and A. Granville, On the equations $z^m = F(x,y)$ and $Ax^p + By^q = Cz^r$, Bull. London Math. Soc. 27 (1995), 513-543.
• [6] P. Dénes, Über die Diophantische Gleichung $x^l+y^l = cz^l$, Acta Math. 88 (1952), 241-251.
• [7] F. Diamond, On deformation rings and Hecke rings, Ann. of Math., to appear.
• [8] F. Diamond and K. Kramer, Modularity of a family of elliptic curves, Math. Res. Lett. 2 (1995), 299-304.
• [9] L. E. Dickson, History of the Theory of Numbers, Chelsea, New York, 1971.
• [10] L. E. Dickson, Introduction to the Theory of Numbers, University of Chicago Press, Chicago, 1929.
• [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.
• [12] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Grad. Texts in Math. 84, 2nd ed., Springer, Berlin, 1990
• [13] S. Kamienny, Rational points on Shimura curves over fields of even degree, Math. Ann. 286 (1990), 731-734.
• [14] B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 33-186.
• [15] B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.
• [16] B. Mazur, Questions about number, in: New Directions in Mathematics, to appear.
• [17] F. Momose, Rational points on the modular curves $X_{split}(p)$, Compositio Math. 52 (1984), 115-137.
• [18] K. A. Ribet, On modular representations of $Gal(ℚ̅/ℚ)$ arising from modular forms, Invent. Math. 100 (1990), 431-476.
• [19] J.-P. Serre, Sur les représentations modulaires de degré 2 de $Gal(ℚ̅/ℚ)$, Duke Math. J. 54 (1987), 179-230.
• [20] R. L. Taylor and A. Wiles, Ring theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553-572.
• [21] H. Wasserman, Variations on the exponent-3 Fermat equation, manuscript, 1995.
• [22] A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. 141 (1995), 443-551.
Typ dokumentu
Bibliografia
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