Results 151 to 160 of about 321 (189)

Exponential Diophantine Equations

1986
This is a integrated presentation of the theory of exponential diophantine equations. The authors present, in a clear and unified fashion, applications to exponential diophantine equations and linear recurrence sequences of the Gelfond-Baker theory of linear forms in logarithms of algebraic numbers.
T. N. Shorey, R. Tijdeman
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A purely exponential Diophantine equation in three unknowns

Periodica Mathematica Hungarica, 2021
The authors consider the exponential equation \[(1)\;\;\; a^x+(ab+1)^y=b^z\ \ \text{in } x,y,z\in\mathbb{Z}_{>0},\] where \(a,b\) are integers \(>1\). They give a list of various pairs \((a,b)\) for which (1) is solvable. This list consists of a couple of infinite families and a finite number of isolated cases. With the exception of two of the infinite
Takafumi Miyazaki, Masaki Sudo, Nobuhiro Terai   +2 more
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A Pure Exponential Diophantine Equation

Bulletin of Pure & Applied Sciences- Mathematics and Statistics, 2016
Let 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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TERAI'S CONJECTURE ON EXPONENTIAL DIOPHANTINE EQUATIONS

International Journal of Number Theory, 2011
Let 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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Exponential Diophantine Equations

2019
This 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, 2015
Summary: 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 ...
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Classification of Quantifier Prefixes Over Exponential Diophantine Equations

Mathematical Logic Quarterly, 1986
After 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, Hilbert Levitz, A. J. Wilkie   +2 more
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Some conjectures in the theory of exponential Diophantine equations

Publicationes Mathematicae Debrecen, 2000
The 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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ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT

Bulletin of the Australian Mathematical Society, 2014
AbstractLet $a$ 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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