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1.
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 nN. 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).  相似文献   

2.
Integral complete multipartite graphs   总被引:1,自引:0,他引:1  
A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphs Kp1,p2,…,pr=Ka1·p1,a2·p2,…,as·ps with s=3,4. We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s=4, we give a positive answer to a question of Wang et al. [Integral complete r-partite graphs, Discrete Math. 283 (2004) 231-241]. The problem of the existence of integral complete multipartite graphs Ka1·p1,a2·p2,…,as·ps with arbitrarily large number s remains open.  相似文献   

3.
Let f1,…,fd be an orthogonal basis for the space of cusp forms of even weight 2k on Γ0(N). Let L(fi,s) and L(fi,χ,s) denote the L-function of fi and its twist by a Dirichlet character χ, respectively. In this note, we obtain a “trace formula” for the values at integers m and n with 0<m,n<2k and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisely the value of the ratio L(f,χ,m)/L(f,n) for a Hecke eigenform f.  相似文献   

4.
For r = (r1,…, rd) ∈ ?d the mapping τr:?d →?d given byτr(a1,…,ad) = (a2, …, ad,−⌊r1a1+…+ rdad⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?d there exists an integer k > 0 with τrk(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).  相似文献   

5.
A sequence of prime numbers p1,p2,p3,…, such that pi=2pi−1+? for all i, is called a Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the smallest positive integer such that 2pk+? is composite, then we say the chain has length k. It is conjectured that there are infinitely many Cunningham chains of length k for every positive integer k. A sequence of polynomials f1(x),f2(x),… in Z[x], such that f1(x) has positive leading coefficient, each fi(x) is irreducible in Q[x] and fi(x)=xfi−1(x)+? for all i, is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether ?=1 or −1 respectively. If k is the least positive integer such that fk+1(x) is reducible in Q[x], then we say the chain has length k. In this article, for polynomial Cunningham chains of both kinds, we prove that there are infinitely many chains of length k and, unlike the situation in the integers, that there are infinitely many chains of infinite length, by explicitly giving infinitely many polynomials f1(x), such that fk+1(x) is the only term in the sequence that is reducible.  相似文献   

6.
We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a1,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a1,…,ak)-periodic decomposition if and only if Δa1akf=0, where Δaf(x)=f(x+a)−f(x).  相似文献   

7.
Let a,b and n be positive integers and the set S={x1,…,xn} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that xσ(1)|…|xσ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (Sa) having the ath power (xi,xj)a of the greatest common divisor of xi and xj as its i,j-entry divides the bth power GCD matrix (Sb) in the ring Mn(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (Sa) does not divide the bth power GCD matrix (Sb) in the ring Mn(Z). Similar results are also established for the power LCM matrices.  相似文献   

8.
For a set A of nonnegative integers the representation functions R2(A,n), R3(A,n) are defined as the number of solutions of the equation n=a+a,a,aA with a<a, a?a, respectively. Let D(0)=0 and let D(a) denote the number of ones in the binary representation of a. Let A0 be the set of all nonnegative integers a with even D(a) and A1 be the set of all nonnegative integers a with odd D(a). In this paper we show that (a) if R2(A,n)=R2(N?A,n) for all n?2N−1, then R2(A,n)=R2(N?A,n)?1 for all n?12N2−10N−2 except for A=A0 or A=A1; (b) if R3(A,n)=R3(N?A,n) for all n?2N−1, then R3(A,n)=R3(N?A,n)?1 for all n?12N2+2N. Several problems are posed in this paper.  相似文献   

9.
Let m be a positive integer and fm(x) be a polynomial of the form fm(x)=x2+xm. We call a polynomial fm(x) a Rabinowitsch polynomial if for and consecutive integers x=x0,x0+1,…,x0+s−1, |fm(x)| is either 1 or prime. In this paper, we show that there are exactly 14 Rabinowitsch polynomials fm(x).  相似文献   

10.
Let r be a positive integer and f1,…,fr be distinct polynomials in Z[X]. If f1(n),…,fr(n) are all prime for infinitely many n, then it is necessary that the polynomials fi are irreducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f1(n)···fr(n), as n runs over Z. Assuming these necessary conditions, Bateman and Horn (Math. Comput.16 (1962), 363-367) proposed a conjectural asymptotic estimate on the number of positive integers n?x such that f1(n),…,fr(n) are all primes. In the present paper, we apply the Hardy-Littlewood circle method to study the Bateman-Horn conjecture when r?2. We consider the Bateman-Horn conjecture for the polynomials in any partition {f1,…,fs}, {fs+1,…,fr} with a linear change of variables. Our main result is as follows: If the Bateman-Horn conjecture on such a partition and change of variables holds true with some conjectural error terms, then the Bateman-Horn conjecture for f1,…,fr is equivalent to a plausible error term conjecture for the minor arcs in the circle method.  相似文献   

11.
We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetic progressions and lower bounds less efficient (but nontrivial) for quadratic sequences whose general term has the form un=an(n+t)+b with (a,t,b)∈Z3, a?5, t?0, gcd(a,b)=1. From this, we deduce for instance the lower bound: lcm{12+1,22+1,…,n2+1}?0,32n(1,442) (for all n?1). In the last part of this article, we study the integer lcm(n,n+1,…,n+k) (kN, nN). We show that it has a divisor dn,k simple in its dependence on n and k, and a multiple mn,k also simple in its dependence on n. In addition, we prove that both equalities: lcm(n,n+1,…,n+k)=dn,k and lcm(n,n+1,…,n+k)=mn,k hold for an infinitely many pairs (n,k).  相似文献   

12.
Given a continued fraction [a0;a1,a2,…], pn/qn=[a0;a1,…,an] is called the n-th convergent for n=0,1,2,…. Leaping convergents are those of every r-th convergent prn+i/qrn+i (n=0,1,2,…) for fixed integers r and i with r?2 and i=0,1,…,r-1. This leaping step r can be chosen as the length of period in the continued fraction. Elsner studied the leaping convergents p3n+1/q3n+1 for the continued fraction of and obtained some arithmetic properties. Komatsu studied those p3n/q3n for (s?2). He has also extended such results for some more general continued fractions. Such concepts have been generalized in the case of regular continued fractions. In this paper leaping convergents in the non-regular continued fractions are considered so that a more general three term relation is satisfied. Moreover, the leaping step r need not necessarily to equal the length of period. As one of applications a new recurrence formula for leaping convergents of Apery’s continued fraction of ζ(3) is shown.  相似文献   

13.
Ramsey regions     
Let (T1,T2,…,Tc) be a fixed c-tuple of sets of graphs (i.e. each Ti is a set of graphs). Let R(c,n,(T1,T2,…,Tc)) denote the set of all n-tuples, (a1,a2,…,an), such that every c-coloring of the edges of the complete multipartite graph, Ka1,a2,…,an, forces a monochromatic subgraph of color i from the set Ti (for at least one i). If N denotes the set of non-negative integers, then R(c,n,(T1,T2,…,Tc))⊆Nn. We call such a subset of Nn a “Ramsey region”. An application of Ramsey's Theorem shows that R(c,n,(T1,T2,…,Tc)) is non-empty for n?0. For a given c-tuple, (T1,T2,…,Tc), known results in Ramsey theory help identify values of n for which the associated Ramsey regions are non-empty and help establish specific points that are in such Ramsey regions. In this paper, we develop the basic theory and some of the underlying algebraic structure governing these regions.  相似文献   

14.
Let G be a graph and a1,…,ar be positive integers. The symbol G→(a1,…,ar) denotes that in every r-coloring of the vertex set V(G) there exists a monochromatic ai-clique of color i for some i∈{1,…,r}. The vertex Folkman numbers F(a1,…,ar;q)=min{|V(G)|:G→(a1,…,ar) and Kq?G} are considered. Let ai, bi, ci, i∈{1,…,r}, s, t be positive integers and ci=aibi, 1?ai?s,1?bi?t. Then we prove that
F(c1,c2,…,cr;st+1)?F(a1,a2,…,ar;s+1)F(b1,b2,…,br;t+1).  相似文献   

15.
Consider the multiplicities ep1(n),ep2(n),…,epk(n) in which the primes p1,p2,…,pk appear in the factorization of n!. We show that these multiplicities are jointly uniformly distributed modulo (m1,m2,…,mk) for any fixed integers m1,m2,…,mk, thus improving a result of Luca and St?nic? [F. Luca, P. St?nic?, On the prime power factorization of n!, J. Number Theory 102 (2003) 298-305]. To prove the theorem, we obtain a result regarding the joint distribution of several completely q-additive functions, which seems to be of independent interest.  相似文献   

16.
The paper is devoted to the study of fractal properties of subsets of the set of non-normal numbers with respect to Rényi f  -expansions generated by continuous increasing piecewise linear functions defined on [0,+∞)[0,+). All such expansions are expansions for real numbers generated by infinite linear IFS f={f0,f1,…,fn,…}f={f0,f1,,fn,} with the following list of ratios Q=(q0,q1,…,qn,…)Q=(q0,q1,,qn,).  相似文献   

17.
Let Un ⊂ Cn[ab] be an extended Chebyshev space of dimension n + 1. Suppose that f0 ∈ Un is strictly positive and f1 ∈ Un has the property that f1/f0 is strictly increasing. We search for conditions ensuring the existence of points t0, …, tn ∈ [ab] and positive coefficients α0, …, αn such that for all f ∈ C[ab], the operator Bn:C[ab] → Un defined by satisfies Bnf0 = f0 and Bnf1 = f1. Here it is assumed that pn,k, k = 0, …, n, is a Bernstein basis, defined by the property that each pn,k has a zero of order k at a and a zero of order n − k at b.  相似文献   

18.
Let K be a number field, and let W be a subspace of KN, N?1. Let V1,…,VM be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in , providing an explicit upper bound on the height of such a point in terms of heights of W and V1,…,VM. Our main tool is a counting estimate we prove for the number of points of a subspace of KN inside of an adelic cube. As corollaries to our main result we derive an explicit bound on the height of a nonvanishing point for a decomposable form and an effective subspace extension lemma.  相似文献   

19.
We devise an efficient algorithm that, given points z1,…,zk in the open unit disk D and a set of complex numbers {fi,0,fi,1,…,fi,ni−1} assigned to each zi, produces a rational function f with a single (multiple) pole in D, such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at zi has fi,0,fi,1,…,fi,ni−1 as its first ni coefficients.  相似文献   

20.
We prove a value distribution result which has several interesting corollaries. Let kN, let αC and let f be a transcendental entire function with order less than 1/2. Then for every nonconstant entire function g, we have that (fg)(k)α has infinitely many zeros. This result also holds when k=1, for every transcendental entire function g. We also prove the following result for normal families. Let kN, let f be a transcendental entire function with ρ(f)<1/k, and let a0,…,ak−1,a be analytic functions in a domain Ω. Then the family of analytic functions g such that
  相似文献   

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