Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant W_n whose (i,j)-th entry is the binomial coefficient , for 1?i,j?n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then and . If q is another prime, distinct from p, and h any positive integer, then . Furthermore, if p is odd, then . In particular, if p?5, then . Also, if m and n are relatively prime positive integers, then W_mW_n divides W_mn.     

A binary linear recurrence sequence of composite numbers

Let (a,b)∈Z², where b≠0 and (a,b)≠(±2,−1). We prove that then there exist two positive relatively prime composite integers x₁, x₂ such that the sequence given by xn+1=axn+bxn−1, n=2,3,… , consists of composite terms only, i.e., |xn| is a composite integer for each n∈N. In the proof of this result we use certain covering systems, divisibility sequences and, for some special pairs (a,±1), computer calculations. The paper is motivated by a result of Graham who proved this theorem in the special case of the Fibonacci-like sequence, where (a,b)=(1,1).     

An application of Fourier transforms on finite Abelian groups to an enumeration arising from the Josephus problem

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We analyze an enumeration associated with the Josephus problem by applying a Fourier transform to a multivariate generating function. This yields a formula for the enumeration that reduces to a simple expression under a condition we call local prime abundance. Under this widely held condition, we prove (Corollary 3.4) that the proportion of Josephus permutations in the symmetric group Sn that map t to k (independent of the choice of t and k) is 1/n. Local prime abundance is intimately connected with a well-known result of S.S. Pillai, which we exploit for the purpose of determining when it holds and when it fails to hold. We pursue the first case where it fails, reducing an intractable DFT computation of the enumeration to a tractable one. A resulting computation shows that the enumeration is nontrivial for this case.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=DnZi-Znuk-A.     

On small solutions to quadratic congruences

We estimate the deviation of the number of solutions of the congruence

     

Jacobi continued fraction and Hankel determinants of the Thue-Morse sequence

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.     

Representing powers of 2 by a sum of four integers

We solve an arithmetic problem due to Erdös and Freud (1986) investigated also by Freiman, Nathanson and Sárközy: How many elements from a given set of integers one must take to represent a power of 2 by their sum?     

On divisibility properties of certain multinomial coefficients—II

A recent conjecture of Myerson and Sander concerns divisibility properties of certain multinomial coefficients. We obtain results in this direction by further pursuing a line of attack developed earlier by the first author.     

A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if l,m,n∈{0,1,2,…} then

     

The generalized order-k Fibonacci–Pell sequence by matrix methods

In this paper, we consider the usual and generalized order-k Fibonacci and Pell recurrences, then we define a new recurrence, which we call generalized order-k F–P sequence. Also we present a systematic investigation of the generalized order-k F–P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order-k F–P sequence by matrix methods. Further, we give the generating function and combinatorial representations of these numbers. Also we present an algorithm for computing the sums of the generalized order-k Pell numbers, as well as the Pell numbers themselves.     

A multiple character sum and a multipleL-function

In this paper we introduce a certain sum by using Dirichlet characters. We investigate the Riesz mean of the sum using the analytic properties of the MultipleL-function systematically. Eine überarbeitete Fassung ging am 6. 8. 2001 ein     

Maximum versus minimum: Two properties of the Fibonacci sequence

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.     

On the multiplicity of linear recurrence sequences

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.     

Cubic and quartic congruences modulo a prime

Let p>3 be a prime, and denote the number of solutions of the congruence . In this paper, using the third-order recurring sequences we determine the values of N_p(x³+a₁x²+a₂x+a₃) and N_p(x⁴+ax²+bx+c), and construct the solutions of the corresponding congruences, where a₁,a₂,a₃,a,b,c are integers.     

Complete solution of the polynomial version of a problem of Diophantus

In this paper, we prove that if {a,b,c,d} is a set of four non-zero polynomials with integer coefficients, not all constant, such that the product of any two of its distinct elements plus 1 is a square of a polynomial with integer coefficients, then

(a+b−c−d)2=4(ab+1)(cd+1).     

On integers of the forms k−2 and k2+1

In this paper we prove that if (r,12)?3, then the set of positive odd integers k such that k^r−2ⁿ has at least two distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to k^r2ⁿ+1 is also true.     

On random perturbation of some intersective sets

Given a subset S of Z and a sequence I = (I_n)_n=1^∞ of intervals of increasing length contained in Z, let

     

On the Function w( x)=|{1= s= k : x= a s (mod n s)}|

For a finite system of arithmetic sequences the covering function is w(x) = |{1 s k : x a_s (mod n_s)}|. Using equalities involving roots of unity we characterize those systems with a fixed covering function w(x). From the characterization we reveal some connections between a period n₀ of w(x) and the moduli n₁, . . . , n_k in such a system A. Here are three central results: (a) For each r=0,1, . . .,n_k/(n₀,n_k)–1 there exists a Jc{1, . . . , k–1} such that . (b) If n₁ ···n_k–l <n_k–l+1 =···=n_k (0 < l < k), then for any positive integer r < n_k/n_k–l with r 0 (mod n_k/(n₀,n_k)), the binomial coefficient can be written as the sum of some (not necessarily distinct) prime divisors of n_k. (c) max_(xw(x) can be written in the form where m₁, . . .,m_k are positive integers.The research is supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.     

On zero-sum sequences of prescribed length

Summary. Let k ≥ 1 be any integer. Let G be a finite abelian group of exponent n. Let s_k(G) be the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length kn. We study this constant for groups when d = 3 or 4. In particular, we prove, as a main result, that for every k ≥ 4, and for every prime p ≥ 5.