摘要 :
The aim of the present article is to give a complete list of the integer solutions to polynomial-exponential Diophantine equations, as well as those to exponential Diophantine equations in three variables. Since there is no univer...
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The aim of the present article is to give a complete list of the integer solutions to polynomial-exponential Diophantine equations, as well as those to exponential Diophantine equations in three variables. Since there is no universal effective method to determine the integer solutions to these equations, our results give new examples in the cases where the theory of linear forms in logarithms cannot be applied.
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摘要 :
Let a, b, c, x and y be positive integers. In this paper we sharpen a result of Le by showing that the
Diophantine equation
axm byn
D c; gcd.ax; by/ D 1
has at most two positive integer solutions .m; n/ satisfying min.m; n/ > 1.
摘要 :
Let a, b, c, x and y be positive integers. In this paper we sharpen a result of Le by showing that the Diophantine equation ax~m-by~n=c; gcd(ax, by)= 1 has at most two positive integer solutions(m; n) satisfying min(m; n)> 1.
摘要 :
In this note, we find all the solutions of the Diophantine equation x~2 + 2~a · 5~b = y~n in positive integers x, y, a, b, n with x and y coprime and n ≥ 3.
摘要 :
Let N be a fixed positive integer and f : R -> C. As a generalisation of the superstability of the exponential functional equation we consider the functional inequalities
摘要 :
In this paper, we prove the equation in the title has no positive integer solutions (x, y, n) with 2 X n and x not equal y apart from (x, y, n) = (5, 2, 5) (90, 2, 13). (c) 2005 Elsevier Inc. All rights reserved.
摘要 :
Let b be a fixed positive integer with b >2. In this paper, using someelementary methods, we prove that if 3| b, then the equation (2~n — 1)(b~n — 1) =x~2hasno positive integer solution (n, x).
摘要 :
A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable x and the temporal variable t, and they are expo...
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A systematic method is presented to provide various equivalent solution formulas for exact solutions to the sine-Gordon equation. Such solutions are analytic in the spatial variable x and the temporal variable t, and they are exponentially asymptotic to integer multiples of 2π as x →±∞. The solution formulas are expressed explicitly in terms of a real triplet of constant matrices. The method presented is generalizable to other integrable evolution equations where the inverse scattering transform is applied via the use of aMarchenko integral equation. By expressing the kernel of that Marchenko equation using a matrix exponential in terms of the matrix triplet and by exploiting the separability of that kernel, an exact solution formula to the Marchenko equation is derived, yielding various equivalent exact solution formulas for the sine-Gordon equation.
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