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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


IMPACT FACTOR 2018: 0.490

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1337-2211
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Volume 69, Issue 2

Issues

The generalized Fermat Conjecture

Adalberto García-Máynez
  • Instituto de Matemáticas, Universidad Nacional Autónoma de México Area de la Investigación Científica Circuito Exterior Ciudad Universitaria, Coyoacán, 04510, Mexico
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/ Margarita Gary
  • Departamento de Ciencias Naturales y Exactas, Universidad de la Costa-CUC, calle 58 # 55-66, Barranquilla, Colombia
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/ Adolfo Pimienta Acosta
  • Facultad de Ciencias Básicas y Biomédicas, Universidad Simón Bolívar, calle 58 # 55-132, Barranquilla, Colombia
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Published Online: 2019-03-18 | DOI: https://doi.org/10.1515/ms-2017-0225

Abstract

If a, b, c are non-zero integers, we considerer the following problem: for which values of n the line ax + by + cz = 0 may be tangent to the curve xn + yn = zn?

We give a partial solution: if n = 5 or if n – 1 is a prime a number, then the answer is the line cannot be tangent to the curve. This problem is strongly related to Fermat’ s Last Theorem.

MSC 2010: Primary 11R04; 11R09; 11R11; 11R58; Secondary 41A50

Keywords: tangent; Fermat curve; Chebyshev polynomials

The first and third author thank to Dr. Francisco González Acuña for several fruitful conversations about the solution of the conjecture in the title.

The second author was supported by the Universidad de la Costa (www.cuc.edu.co): Departamento de Ciencias Naturales y Exactas and by Grupo de Investigación en Ciencias Naturales y Exactas, GICNEX.

The third author was supported by Universidad Simón Bolívar (www.unisimon.edu.co) under grant of the Faculty of Basic Sciences, Barranquilla, Colombia.

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    Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Math., Springer-Verlag, New York, 1996.Google Scholar

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    Wiles, A.: Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443–551.CrossrefGoogle Scholar

About the article

Received: 2017-07-23

Accepted: 2018-05-05

Published Online: 2019-03-18

Published in Print: 2019-04-24


(Communicated by Filippo Nuccio)


Citation Information: Mathematica Slovaca, Volume 69, Issue 2, Pages 321–326, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0225.

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