A certain alternating sum u(n) of n + 1 products of two binomial coefficients has a property similar to Wolstenholme’s theorem, namely u(p) ≡ −1 (mod p3) for all primes p ≥ 5. However, this congruence also holds for certain composite integers p which appear to always have exactly two prime divisors, one of which is always 2 or 5. This phenomenon will be partly explained and the composites in qu...

Let Fq be a finite field of odd order q. In this paper, the irreducible factorization of x2 pr − 1 over Fq is given in a very explicit form, where a, b, c are positive integers and p, r are odd prime divisors of q−1. It is shown that all the irreducible factors of x2 pr − 1 over Fq are either binomials or trinomials. In general, denote by vp(m) the degree of prime p in the standard decompositio...

Let L be a non empty loop structure. We say that L is add-left-cancelable if and only if: (Def. 1) For all elements a, b, c of L such that a + b = a + c holds b = c. We say that L is add-right-cancelable if and only if: (Def. 2) For all elements a, b, c of L such that b + a = c + a holds b = c. We say that L is add-cancelable if and only if: (Def. 3) For all elements a, b, c of L holds if a + b...

We study a polynomial sequence Cn(x|q) defined as a solution of a q-difference equation. This sequence, evaluated at q-integers, interpolates Carlitz–Riordan’s q-ballot numbers. In the basis given by some kind of q-binomial coefficients, the coefficients are again some qballot numbers. We obtain another curious recurrence relation for these polynomials in a combinatorial way.

We prove a general theorem about the multiplicity of the entries in certain integer arrays which is best possible in general. As an application we give non-trivial bounds for the multiplicities of several well-known combinatorial arrays including the binomial coefficients, Narayana numbers and the Eulerian numbers. For the binomial coefficients we obtain the result of Singmaster.

We first extend the digital binomial identity as given by Nguyen et al. to an identity in an arbitrary base b, by introducing the b-ary binomial coefficients. Then, we study the properties of these coefficients such as their orthogonality, their link with Lucas’ theorem and their extension to multinomial coefficients. Finally, we analyze the structure of the corresponding b-ary Pascal-like tria...

Abstract This paper proposes a professional English translation corpus based on the binomial theorem coefficients. combines with vector space model to analyze similarity between machine and human in translation. The study results show that both can experience inaccuracies. is more objective accurate. established by this method help us solve problem of terminology research has specific reference...

We study the degree of nonhomogeneous lattice ideals over arbitrary fields, and give formulas to compute the degree in terms of the torsion of certain factor groups of Zs and in terms of relative volumes of lattice polytopes. We also study primary decompositions of lattice ideals over an arbitrary field using the Eisenbud–Sturmfels theory of binomial ideals over algebraically closed fields. We ...

We prove the divisibility conjecture on sums of even powers q-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies two q-harmonic series congruences due to Shi Pan.