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Solution of the Exponential Diophantine Equation

Journal of Chemical, Biological and physical sciences
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On exponential Diophantine equations concerning Pythagorean triples

Publicationes Mathematicae Debrecen, 2022
As 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

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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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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On Some Exponential Diophantine Equations

Monatshefte f�r Mathematik, 2001
Let \(D_1,D_2\) be coprime positive integers, and let \(h\) denote the class number of the quadratic field \(\mathbb{Q} (\sqrt{-D_1D_2})\). In this paper, using a deep theorem concerning the existence of primitive divisors of Lucas and Lehmer numbers, the authors completely determine all solutions \((x,y,n)\) of the generalized Ramanujan-Nagell ...
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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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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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