Results 111 to 120 of about 7,521 (156)

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 ...
Yann Bugeaud
openaire   +3 more sources

Conjunctions of exponential diophantine equations over $${\mathbb {Q}}$$

Archive for Mathematical Logic
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
openaire   +3 more sources

The Decision Problem for Exponential Diophantine Equations

The Annals of Mathematics, 1961
Davis, Martin, Putnam, Hilary, Robinson, Julia   +2 more
openaire   +4 more sources

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
openaire   +2 more sources

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
openaire   +1 more source

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
openaire   +1 more source

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
openaire   +1 more source

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 \(x, y\).
openaire   +1 more source

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
openaire   +1 more source