1. On a conjecture on exponential Diophantine equations
- Author
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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
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