Results 141 to 150 of about 312 (174)

An Exponential Diophantine Equation: 10873

The American Mathematical Monthly, 2003
Solution 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, 2023
Summary: 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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ON THE EXPONENTIAL DIOPHANTINE EQUATION

Bulletin of the Australian Mathematical Society, 2014
AbstractLet$m$,$a$,$c$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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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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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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Solving Exponential Diophantine Equations

Mathematics Magazine, 1981
Leo J. Alex
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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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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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Some exponential Diophantine equations

, 1988
T. N. Shorey
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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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