共查询到20条相似文献,搜索用时 31 毫秒
1.
Hao Pan 《Discrete Mathematics》2006,306(16):1921-1940
By a very simple argument, we prove that if l,m,n∈{0,1,2,…} then
2.
The sequence {xn} defined by xn=(n+xn−1)/(1−nxn−1), with x1=1, appeared in the context of some arctangent sums. We establish the fact that xn≠0 for n?4 and conjecture that xn is not an integer for n?5. This conjecture is given a combinatorial interpretation in terms of Stirling numbers via the elementary symmetric functions. The problem features linkage with a well-known conjecture on the existence of infinitely many primes of the form n2+1, as well as our conjecture that (1+12)(1+22)?(1+n2) is not a square for n>3. We present an algorithm that verifies the latter for n?103200. 相似文献
4.
Patrick B. Allen 《Journal of Number Theory》2007,126(2):212-216
We prove a lemma regarding the linear independence of certain vectors and use it to improve on a bound due to Schmidt on the zero-multiplicity of linear recurrence sequences. 相似文献
5.
In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))n?0 and (U(n))n?0, which are ultimately periodic modulo m, e.g.
6.
Guo-Niu Han 《Quaestiones Mathematicae》2016,39(7):895-909
We study the Jacobi continued fraction and the Hankel determinants of the Thue-Morse sequence and obtain several interesting properties. In particular, a formal power series φ(x) is being discovered, having the property that the Hankel transforms of φ(x) and of φ(x2) are identical. 相似文献
7.
Hans Peter Schlickewei Wolfgang M. Schmidt Michel Waldschmidt 《manuscripta mathematica》1999,98(2):225-241
Let be an exponential polynomial over a field of zero characteristic. Assume that for each pair i,j with i≠j, α
i
/α
j
is not a root of unity. Define . We introduce a partition of into subsets (1≤i≤m), which induces a decomposition of f into , so that, for 1≤i≤m, , while for , the number either is transcendental or else is algebraic with not too small a height. Then we show that for all but at most solutions x∈ℤ of f(x)= 0, we have
Received: 7 August 1998 相似文献
8.
9.
Hao Pan 《Journal of Number Theory》2008,128(6):1646-1654
Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define
10.
Zhi-Wei Sun 《Journal of Number Theory》2011,131(12):2387-2397
The nth Delannoy number and the nth Schröder number given by
11.
Zhi-Hong Sun 《Journal of Number Theory》2008,128(2):280-312
Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have
12.
It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci∈[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type. 相似文献
13.
14.
Zhi-Wei Sun 《Journal of Number Theory》2005,115(2):371-380
To determine Euler numbers modulo powers of two seems to be a difficult task. In this paper we achieve this and apply the explicit congruence to give a new proof of a classical result due to M.A. Stern. 相似文献
15.
The well-known Favard's theorem states that the linear differential equation
(1) 相似文献
16.
Guodong Liu 《Journal of Number Theory》2008,128(12):3063-3071
In this paper, we establish some identities involving the Euler numbers, the Euler numbers of order 2 and the central factorial numbers, and give a new proof of a classical result due to M.A. Stern.
Video abstract
For a video summary of this paper, please visit http://www.youtube.com/watch?v=kdNsdTDA-FE. 相似文献17.
It is well-known that the Fibonacci numbers have a maximum property with respect to the length of the regular continued fraction expansion (or, equivalently, of the Euclidean algorithm). But it seems to be scarcely known that they also have a minimum property relative to the sum of the digits of this expansion. We discuss both properties and their interrelation here. 相似文献
19.
Zhi-Hong Sun 《Journal of Number Theory》2007,124(1):62-104
Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u2−dv2 and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms. 相似文献
20.
Summary The aim of this paper is to generalize the well-known Eulerian numbers, defined by the recursion relationE(n, k) = (k + 1)E(n – 1, k) + (n – k)E(n – 1, k – 1), to the case thatn is replaced by . It is shown that these Eulerian functionsE(, k), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. TheE(, k) satisfy recursion formulae, they are monotone ink and, as functions of , are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, theS(, k), > 0, introduced by the authors (1989), are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation ofx
in terms of a sum involving theE(, k) and a hypergeometric function.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth. 相似文献