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TERAI'S CONJECTURE ON EXPONENTIAL DIOPHANTINE EQUATIONS
International Journal of Number Theory, 2011Let a, b, c be relatively prime positive integers such that ap + bq = cr with fixed integers p, q, r ≥ 2. Terai conjectured that the equation ax + by = cz has no positive integral solutions other than (x, y, z) = (p, q, r) except for specific cases. Most known results on this conjecture concern the case where p = q = 2 and either r = 2 or odd r ≥3. In
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A Pure Exponential Diophantine Equation
Bulletin of Pure & Applied Sciences- Mathematics and Statistics, 2016Let u be an even integer. In this paper we prove that if 4| u and u ≥208, then the equation (u3-3u)x + (3u2-1)y = (u2 +1)z has only the positive integer solution (x, y, z) = (2, 2, 3). This result basically determine all solutions of the equation.
Zhong Li, Wei-xun Li
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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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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 +4 more sources
Classification of Quantifier Prefixes Over Exponential Diophantine Equations
Mathematical Logic Quarterly, 1986After Matijasevič had solved the \(10^{th}\) problem of Hilbert in 1970, it was natural to consider other similar problems. The \(10^{th}\) problem of Hilbert can be viewed as the problem of classification of the quantifier prefix \(\exists\exists\ldots\exists\) over diophantine equations (polynomial equations).
James P. Jones +2 more
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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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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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Exponential Diophantine equations over function fields
Publicationes Mathematicae Debrecen, 1992Let \(k\) be an algebraically closed field of characteristic zero and let \(k(t)\) be the field of rational functions over \(k\). Further, let \(\mathbb{K}\) be a finite extension of \(k(t)\). For given non-zero elements \(f_ 1,\ldots,f_ n,g\) of \(\mathbb{K}[X_ 1,\ldots,X_ n]\) \((n\geq 2)\), consider the equation \[ \sum^ n_{i=1}f_ i({\mathbf x ...
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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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