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On Some Exponential Diophantine Equations 总被引:10,自引:0,他引:10
Yann Bugeaud 《Monatshefte für Mathematik》2001,132(2):93-97
We apply a new, deep result of Bilu, Hanrot and Voutier to solve completely some exponential Diophantine equations of the type , where are given coprime positive integers, , and are unknown. (Received 29 May 2000; in revised form 21 September 2000) 相似文献
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关于指数型丢番图方程的整数解 总被引:4,自引:0,他引:4
本文简要地介绍了有关S-单位方程、Ramanujan-Nagell方程、Thue-Mahler方程、LeVeque方程、Catalan方程、Pillai方程等指数型丢番图方程整数解的最新结果。这些结果大多是用Thue-Siegle-Roth-Schmidt方法和Gel’fond-Baker方法得到的。 相似文献
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本文首先证明了一些丢番图方程在整数环Z中无解,然后我们部分验证了Browkin在[Lecture Notes in Math.,Vol.966,1-6]中的一个猜想. 相似文献
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本文研究了和素数有关的方程组和不等式组,通过更细致的指数和估计,我们改进了以前的结果。 相似文献
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Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the
first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of
over
for all
.
The first author was supported by the Austrian Science Foundation, grants S8307-MAT and J2407-N12.
The second author was supported by the Austrian Science Foundation, grant S8307-MAT. 相似文献
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Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the
first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of
X(X-Y)(X-(T+x)Y)+Y3=1+xT(1-T)X(X-Y)(X-(T+\xi)Y)+Y^3=1+\xi T(1-T)
over
\Bbb C[T]{\Bbb C}[T]
for all
x ? \Bbb C\xi\in{\Bbb C}
. 相似文献
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Yu. F. Bilu B. Brindza P. Kirschenhofer Á. Pintér R. F. Tichy A. Schinzel 《Compositio Mathematica》2002,131(2):173-188
Given m, n 2, we prove that, for sufficiently large y, the sum 1
n
+···+ y
n
is not a product of m consecutive integers. We also prove that for m n we have 1
m
+···+ x
m
1
n
+···+ y
n
, provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are almost indecomposable, a result of independent interest. 相似文献
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We consider difference equations of the form $$x_{n+1}={x^k_n+aover x^l_nx_{n-1}}$$ 相似文献
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We investigate the behavior of the positive solutions of a class of difference equations with strong negative feedback. In particular we show that the positive equilibrium is globally asymptotically stable. 相似文献
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高凌云 《数学年刊A辑(中文版)》2014,35(2):193-202
研究了具有允许的亚纯解的复差分方程的形式以及系数的级与解的级两者的关系,得到了两个结果.将复微分方程中一些结果推广至复差分方程. 相似文献
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We consider quadratic diophantine equations of the shape
for a polynomial Q(X1, ..., Xs) Z[X1, ..., Xs] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound s(H) with the following property: if equation (1) hasa solution x Zs at all, then it has one satisfying
For s = 3 and s = 4 no polynomial bounds s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55. 相似文献
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Rong Fang BIE Shi Qiang WANG 《数学学报(英文版)》2006,22(5):1549-1556
In this paper, by using model-theoretic methods, it is shown that some systems of unsolved cubic diophantine equations in number theory can have solutions in certain inductive extension rings of the ring I of rational integers. These inductive rings are not fields, and every element of them is a sum of 4 cubes and a sum of 3 squares. Also some of them satisfy the Goldbach conjecture and some others don't. 相似文献
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This paper proves conditional existence results for non-trivialsolutions of the equation where the coefficients ai and the unknowns Xi are taken to berational integers. No such results were previously known for n6. The proofs useelementary facts about the 3-descent procedure for ellipticcurves of the form EA: X3 + Y3 = AZ3. Thus, when n=4, and the ai are each prime, and are all congruentto 2 modulo 3, it is shown that (*) will have non-trivial solutions,providing that the Selmer conjecture holds for the curves EA.One may replace the Selmer conjecture by an appropriate formof the Generalized Riemann Hypothesis, when n=5 and the ai areagain taken to be primes, all congruent to 8 modulo 9. Finally,when n=5, one may require only that the ai be square-free andcoprime to 3, providing one assumes both the Selmer conjectureand a special case of Schinzel's conjecture (on the representationof primes by cubic polynomials). 1991 Mathematics Subject Classification:11D25, 11G05, 14G05. 相似文献
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S. S. Pillai 《Proceedings Mathematical Sciences》1940,12(2):199-201
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H.D. Voulov 《Journal of Difference Equations and Applications》2013,19(9):799-810
We consider positive solutions of the following difference equation x n =max A x n m k , B x n m m , n =0,1,…, where A , B are any positive real numbers and k , m are any positive integers. We prove that every positive solution is eventually periodic and determine the period in terms of the parameters A , B , k , and m . 相似文献
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This paper deals with the solvability of initial-value problems (IVPs) and multipoint boundary-value problems (MPBVPs) for linear implicit non-autonomous systems of difference equations. 相似文献
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Weili Yao 《The Ramanujan Journal》2017,44(1):155-175
Let \(\Omega (n)\) be the total number of prime factors of n, and let \(\lambda _j\) be the real numbers satisfying suitable conditions. Let \(J_k(N)\) denote the number of solutions to the inequality In this note, we investigate the properties of \(J_k(N)\) for any integer \(k\ge 1\), which is allowed to tend to infinity with respect to N. Using an asymptotic formula for the weighted exponential sums, we obtain a sharper lower bound for it and also discuss an application of the main result.
相似文献
$$\begin{aligned} \left\{ \begin{array}{ll} |\lambda _1n_1+\lambda _2n_2+\lambda _3n_3+\eta |<\varepsilon ,&{} \\ \Omega (n_j)= k,&{} \\ 2\le n_j\le N \quad (j=1,2,3).&{} \end{array} \right. \end{aligned}$$