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On Some Exponential Diophantine Equations [PDF]

Monatshefte Fur 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 ...
Yann Bugeaud, Bugeaud Yann
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On the Exponential Diophantine Equation $$(m^2+m+1)^x+m^y=(m+1)^z $$ [PDF]

Mediterranean Journal of Mathematics, 2020
Let m >= 1 be a positive integer. We show that the exponential Diophantine equation (m(2) + m + 1)(x) + m(y) = (m + 1)(z) has no positive integer solutions other than (x, y, z) = (1, 1, 2) when m is not an element of {1, 2, 3}
Murat Alan
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On the solutions of the exponential Diophantine equation ax+by= (m2+ 1)z [PDF]

Quaestiones Mathematicae, 2013
Click on the link to view the abstract.Keywords: Exponential Diophantine equation, Terai conjecture, positive integer solution, linear forms in two logarithms, lower bound, primitive divisorQuaestiones Mathematicae 36(2013), 119 ...
, Alain Togbé
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ON THE SOLUTION OF A CLASS OF EXPONENTIAL DIOPHANTINE EQUATIONS

South East Asian J. of Mathematics and Mathematical Sciences, 2022
In this note, we show that for n = 4N + 3, N     N      0  , the expo- nential Diophantine equation nx + 24y = z2 has exactly two solutions if n + 1 or equivalently N + 1 is an square. When N + 1 = m2, the solutions are given by (0, 1, 5) and (1, 0, 2m). Otherwise it has a unique solution (0, 1, 5) in non-negative integers.
Dutta, Mridul, Borah, Padma Bhushan
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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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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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