关于Fibonacci三角形猜想k=6的证明

运用初等方法,证明k=6时Fibonacci三角形不存在.     

On recurrences for sums of powers of binomial coefficients

The recurrence for sums of powers of binomial coefficients is considered and a lower bound for the minimal length of the recurrence is obtained by using the properties of congruence.

Video abstract

For a video summary of this paper, please visit http://www.youtube.com/watch?v=jwy6B4aYR-Q.     

On the solutions of a system of two Diophantine equations

We obtain all positive integer solutions(m1,m2,a,b)with ab,gcd(a,b)=1 to the system of Diophantine equations km21-lat1bt2a2r=C1,km22-lat1bt2b2r=C2,with C1,C2∈{-1,1,-2,2,-4,4},and k,l,t1,t2,r∈Z such that k0,l0,r0,t10,t2 0,gcd(k,l)=1,and k is square-free.     

On some questions of Razpet regarding binomial coefficients

We answer two questions of Razpet (Discrete Math. 135 (1994) 377) regarding finite submatrices of the Pascal triangle. One of these has been solved independently in another way by Bayat and Teimoori (Linear Algebra Appl. 308 (2000) 65).     

On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g., for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions. Received November 5, 1998, Accepted December 7, 2000.     

On some combinatorial formulae coming from Hessian Topology

We prove that the fields of asymptotic lines of a real hyperbolic homogeneous polynomial are isotopic to the corresponding fields of its hyperbolic homogeneous part. We also show some combinatorial identities which are related to such isotopy.     

On a Conjecture of J. Weidmann

Consider the Sturm-Liouville differential expression l(y) = ?y″ +q(x)y on an interval (a,b) and assume that l is in the limit point case at b. Fix c ∈(a,b) and let L, Lb be self-adjoint realizations of l in ?²(a,b), ?²(c,b) respectively. If Lb has purely absolutely continuous spectrum in an interval J and if the spectral function ρ_b of Lb satisfies some mild growth conditions then the spectrum of L in J is shown to be purely absolutely continuous, too. Our result confirms a conjecture of J. Weidmann (1982). It had been shown by del Rio Castillo (1988) that in Weidmann’s original formulation this conjecture is false.     

On a Conjecture of E. Dittert

Let K_n denote the set of all n X n nonnegative matrices whose entries have sum n, and let φ be a real valued function defined on K_n by φ(X) = π_iⁿ⁼¹ n, + π_j=1^c_j⁻ⁿ per X for X E K_n with row sum vector (r₁,…, r_n) and column sum vector (c_l,hellip;, c_n). For the same X, let φ_ij(X)= π_k≠ir_k + π_1≠jc₁ - per X(i| j). A ϵK_n is called a φ-maximizing matrix if φ(A) > φ(X) for all X ϵ K_n. Dittert's conjecture asserts that J_n = [1/n]_n×n is the unique (φ-maximizing matrix on K_n. In this paper, the following are proved: (i) If A = [a_ij] is a φ-maximizing matrix on K_n then φ_ij(A) = φ (A) if a_ij > 0, and φ_ij (A) ⩽ φ (A)if aij = 0. (ii) The conjecture is true for n = 3.     

On a Conjecture of A. Bezdek

The aim of this paper is to prove a conjecture of A. Bezdek. It generalizes conjectures of V. V. Proizvolov. The theorems of this paper are as follows: Suppose points {x _i } _i=1 ^p and a polytope X are given. Then we can choose p facets {F _i } _i=1 ^p of X so that the convex hulls \conv({x _i }\cup F _i ) either cover the polytope or do not intersect pairwise in interior points. Received April 16, 2001, and in revised form October 11, 2001. Online publication March 1, 2002.     

关于丢番图方程xp-1=Dyn

利用分解因子法,讨论了一类高阶丢番图方程的解,并解决了当D=5时,不定方程解的情况.     

Lehmer序列中的平方数与平方类

设Vn(R,Q)表示参数为R和Q的Lehmer伴随序列.如果R和Q为互素奇数且D=R-4Q>0,我们找出了满足Qn(R,Q)或n1Qn(R,Q)是平方数的所有奇数n.这里,从而改进了文[15]的工作.     

On a Conjecture of E. A. Rakhmanov

It is shown that a conjecture of E. A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L _p -norm, 0 < p ≤∈fty , and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discrete complex sets, such as the weighted discrete transfinite diameter and a weighted discrete Bernstein—Walsh-like inequality. August 24, 1998. Date revised: March 26, 1999. Date accepted: April 27, 1999.     

关于Veldman等人的一个猜想

本文证明了下列源于Veldman等人的一个猜想;设G是阶为n≥13的1—tough图且对所有独立集x,y,z有d(x) d(y) d(z)≥(3n-14)/2,则G是哈米顿的。     

Solutions of the congruence

To supplement existing data, solutions of are tabulated for primes with and . For , five new solutions 2^{32}$"> are presented. One of these, for , also satisfies the ``reverse' congruence . An effective procedure for searching for such ``double solutions' is described and applied to the range , . Previous to this, congruences are generally considered for any and fixed prime to see where the smallest prime solution occurs.

     

On a Conjecture of Finotti

We prove a conjecture of Luis Finotti about cubic polynomials of one variable in characteristic p. He checked it by computer for primes p < 890 and uses it to define and study the minimal degree lift of the generic point of an ordinary elliptic curve in characteristic p to the canonical lift mod p ³ of the curve. Received: 5 June 2002     

On a Conjecture of Halperin

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A note on a problem of Brocard-Ramanujan

The Brocard-Ramanujan problem pertains to the Diophantine equation n!+1=m ². Using the quadratic reciprocity law, we prove the existence of many infinite families of positive integers n such that n!+1 is not a perfect square, and we give several simple criteria for n!+1 to be a non-square.      

On a Conjecture of Kemnitz

A classic theorem of Erdis, Ginzburg and Ziv states that in a sequence of 2n-1 integers there is a subsequence of length n whose sum is divisble by n. This result has led to several extensions and generalizations. A multi-dimensional problem from this line of research is the following. Let Z_nZ_n stand for the additive group of integers modulo n. Let s(n, d) denote the smallest integer s such that in any sequence of s elements from Z_n^dZ_n^d (the direct sum of d copies of Z_nZ_n) there is a subsequence of length n whose sum is 0 in Z_n^dZ_n^d. Kemnitz conjectured that s(n, 2) = 4n - 3. In this note we prove that s(p,2) ￡ 4p - 2s(p,2) \le 4p - 2 holds for every prime p. This implies that the value of s(p, 2) is either 4p-3 or 4p-2. For an arbitrary positive integer n it follows that s(n, 2) ￡ (41/10)ns(n, 2) \le (41/10)n. The proof uses an algebraic approach.     

关于高的一个猜想

In this paper, we prove s conjecture formulated by Gao on zero-sum.And also we make an improvement on a result due to Gao.