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Conjunctions of exponential diophantine equations over $${\mathbb {Q}}$$
Archive for Mathematical LogiczbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On exponential Diophantine equations concerning Pythagorean triples
Publicationes Mathematicae Debrecen, 2022As an analogue of the well known Jeśmanovicz conjecture the authors propose to consider the conjecture that the equation \[x^2+(2uv)^m=(u^2+v^2)^n\] has only the positive solutions \((x,m,n)=(u-v,1,1),(u^2-v^2,2,2)\) except for \((u,v)=(244,231)\) and \(3u^2-8uv+3v^2=-1\).\par The authors prove this conjecture for several cases.
Terai, Nobuhiro, Fujita, Yasutsugu
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Exponential Diophantine Equations
1986This 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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ON THE EXPONENTIAL DIOPHANTINE EQUATION
Bulletin of the Australian Mathematical Society, 2014AbstractLet$m$,$a$,$c$be positive integers with$a\equiv 3, 5~({\rm mod} \hspace{0.334em} 8)$. We show that when$1+ c= {a}^{2} $, the exponential Diophantine equation$\mathop{({m}^{2} + 1)}\nolimits ^{x} + \mathop{(c{m}^{2} - 1)}\nolimits ^{y} = \mathop{(am)}\nolimits ^{z} $has only the positive integer solution$(x, y, z)= (1, 1, 2)$under the condition ...
TAKAFUMI MIYAZAKI, NOBUHIRO TERAI
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An Exponential Diophantine Equation: 10873
The American Mathematical Monthly, 2003Solution by Doyle Henderson, Omaha, NE. The only possibilities for m are 0, 1, 2, 3, and 5. Simple calculations then show the solution set for (m, n) to be {(0, 0), (1, 1), (2, 2), (5, 11)1}. Suppose there is a solution with m even and at least 4. Let m = 2a and x = 3a We have x2 2n2 = 1.
B. J. Venkatachala, Doyle Henderson
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An exponential Diophantine equation on triangular numbers
Mathematica Applicanda, 2023Summary: Looking to the two remarkable identities concerning triangular numbers \(T_{n + 1} - T_{n} = n + 1\) and \(T_{n + 1}^{2} - T_{n}^{2} = (n + 1)^{3}\), we can extend these equations to the exponential Diophantine equation \(T_{n + 1}^{x} - T_{n}^{x} = (n + 1)^{y}\) for some positive integers \(x, y\).
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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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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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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).
Jones, J. P., Levitz, H., Wilkie, A. J.
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