The generalized Fermat Conjecture
-
Adalberto García-Máynez
,
Margarita Gary
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.
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.
(Communicated by Filippo Nuccio)
Acknowledgement
The authors would like to thank the referee for his many useful suggestions which have been incorporated in, and have improved the original text, particularly of Section 3.
References
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© 2019 Mathematical Institute Slovak Academy of Sciences
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Articles in the same Issue
- Regular papers
- RNDr. Miloslav Duchoň, DrSc. passed away ∗ aug. 21, 1932 – †dec. 31, 2018
- Prof. RNDr. Gejza Wimmer, DrSc. – A Septuagenerian?
- Two monads on the category of graphs
- Entropy of MV-algebraic dynamical systems: An example
- Going up and lying over in congruence-modular algebras
- The minimal arity of near unanimity polymorphisms
- On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces of real rank one
- The generalized Fermat Conjecture
- Evaluation of sums involving products of Gaussian q-binomial coefficients with applications
- Integrals of logarithmic functions and alternating multiple zeta values
- An abelian subextension of the dyadic division field of a hyperelliptic Jacobian
- The class of all semigroups related to semihypergroups of order 2
- Criterion for the weak potency of certain HNN extensions
- Fekete Szegö theorem for a close-to-convex error function
- Bounds for distance between eigenvalues of boundary value problems with retarded argument
- Gradient estimates for a nonlinear heat equation under the Finsler-Ricci flow
- Weak module amenability of triangular banach algebras II
- Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class
- Example of C-rigid polytopes which are not B-rigid
- Some comments on rU-spaces
- On the equivalence of various definitions of mixed poisson processes
- A note on discrete C-embedded subspaces
