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Your search keyword '"11D09, 11D45, 11J20, 11J86"' showing total 2 results
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2 results on '"11D09, 11D45, 11J20, 11J86"'

1. On a conjecture on exponential Diophantine equations

Author

Cipu, Mihai and Mignotte, Maurice

Subjects
Mathematics - Number Theory, 11D09, 11D45, 11J20, 11J86
Abstract

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.

Published
2008
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2. On a conjecture on exponential Diophantine equations

Author

Maurice Mignotte, Mihai Cipu, 'Simion Stoilow' Institute of Mathematics (IMAR), Romanian Academy of Sciences, Institut de Recherche Mathématique Avancée (IRMA), Université Louis Pasteur - Strasbourg I-Centre National de la Recherche Scientifique (CNRS), The first author has been partially supported by the CEEX Program of the Romanian Ministry of Education, Research and Youth, Grant 2-CEx06-11-20/2006, The Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania, Université Louis Pasteur, U. F. R. de Math\' ematiques, Rue Ren\' e Descartes, 67084 Strasbourg, and France

Subjects
Algebra and Number Theory, Conjecture, Mathematics - Number Theory, exponential equations, Diophantine equation, Of the form, 11D09, 11D45, 11J20, 11J86, [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], Exponential function, Combinatorics, FOS: Mathematics, Number Theory (math.NT), Uniqueness, linear forms in logarithms, Terai's conjecture, Prime power, MSC 11D09, 11D45, 11J20, 11J86, Mathematics
Abstract

We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this is the only solution in integers greater than 1, with the possible exception of finitely many values $(c,r)$. We also prove the uniqueness of such a solution if any of $a$, $b$, $c$ is a prime power. In a different vein, we obtain various inequalities that must be satisfied by the components of a putative second solution.

Published
2009
Full Text
View/download PDF

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