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Exponential Diophantine Equations
2019This paper is a very gentle introduction to solving exponential Diophantine equations using the technology of linear forms in logarithms of algebraic numbers.
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A note on ternary purely exponential diophantine equations
Acta Arithmetica, 2015Summary: Let \(a,b,c\) be fixed coprime positive integers with \(\min\{a,b,c\}>1\), and let \(m=\max \{a,b,c\}\). Using the Gel'fond-Baker method, we prove that all positive integer solutions \((x,y,z)\) of the equation \(a^x+b^y=c^z\) satisfy \(\max \{x,y,z\}1\).
Hu, Yongzhong, Le, Maohua
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ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT
Bulletin of the Australian Mathematical Society, 2014AbstractLet $a$ and $m$ be relatively prime positive integers with $a>1$ and $m>2$. Let ${\it\phi}(m)$ be Euler’s totient function. The quotient $E_{m}(a)=(a^{{\it\phi}(m)}-1)/m$ is called the Euler quotient of $m$ with base $a$. By Euler’s theorem, $E_{m}(a)$ is an integer.
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The Undecidability of Exponential Diophantine Equations
1966Publisher Summary This chapter focuses on the undecidability of exponential Diophantine equations. It is not known whether exponential Diophantine sets are necessarily Diophantine. However, it is known that every exponential Diophantine equation could be transformed mechanically into an equivalent ordinary Diophantine equation in more unknowns ...
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Some conjectures in the theory of exponential Diophantine equations
Publicationes Mathematicae Debrecen, 2000The author formulates a conjecture which implies Pillai's conjecture and a theorem of \textit{A. Schinzel} and \textit{R. Tijdeman} [Acta Arith. 31, 199-264 (1976; Zbl 0339.10018)] that for a polynomial with integer coefficients and at least two distinct roots, there are only finitely many perfect powers in its values at integral points.
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Exponential diophantine equations
Let \(\mathbb G\) be a commutative algebraic group over \(\mathbb C\), not containing any algebraic subgroup isomorphic to the additive group \(\mathbb G_ a\). Let \(\Gamma\) be a subgroup of \(\mathbb G(\mathbb C)\) of finite rank, that is, there is a finitely generated subgroup \(\Gamma'\) of \(\Gamma\) such that all elements of \(\Gamma/\Gamma ...openaire +2 more sources
Integer Solutions of Some Exponential Diophantine Equations
In this paper, we search for non-negative integer solutions to the exponential Diophantine equations. We discussed theorems for their integer solutions.C. Saranya, M. Janani, R. Salini
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On the Exponential Diophantine Equation $$(m^2+m+1)^x+m^y=(m+1)^z $$
Mediterranean Journal of Mathematics, 2020Murat Alan
exaly
On Two Classes of Exponential Diophantine Equations
Communications in Mathematics and Applications, 2022Padma Bhushan Borah, Mridul Dutta
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