Related papers: A Determinant Congruence Conjectured by Sun

In this paper, we prove the following result conjectured by Z.-W. Sun: $$ (2n-1){3n\choose n}| \sum_{k=0}^{n}{6k\choose 3k}{3k\choose k}{6(n-k)\choose 3(n-k)}{3(n-k)\choose n-k}. $$ by showing that the left-hand side divides each summand on…

Let $R$ be an affine algebra over an algebraically closed field of characteristic $0$ with dim$(R)=n$. Let $P$ be a projective $A=R[T_1,\cdots,T_k]$-module of rank $n$ with determinant $L$. Suppose $I$ is an ideal of $A$ of height $n$ such…

Let A=(a_(ij)) be the generic n by n circulant matrix given by a_(ij)=x_(i+j), with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is…

A frequently cited theorem says that for n > 0 and prime p, the sum of the first p n-th powers is congruent to -1 modulo p if p-1 divides n, and to 0 otherwise. We survey the main ingredients in several known proofs. Then we give an…

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$…

In this paper, we establish the following two congruences: \begin{gather*} \sum_{k=0}^{(p+1)/2}(3k-1)\frac{\left(-\frac{1}{2}\right)_k^2\left(\frac{1}{2}\right)_k4^k}{k!^3}\equiv…

We prove that $\det A\leq 6^\frac{n}{6}$ whenever $A\in\{0,1\}^{n\times n}$ contains at most $2n$ ones. We also prove an upper bound on the determinant of matrices with the $k$-consecutive ones property, a generalisation of the consecutive…

Let $A$ and $B$ be complex numbers, and let $(w_n)_{n\ge0}$ be a sequence of complex numbers with $w_{n+1}=Aw_n-Bw_{n-1}$ for all $n=1,2,3,\ldots$. When $w_0=0$ and $w_1=1$, the sequence $(w_n)_{n\ge0}$ is just the Lucas sequence…

We give a family of congruences for the binomial coefficients ${kp-1\choose p-1}$ in terms of multiple harmonic sums, a generalization of the harmonic numbers. Each congruence in this family (which depends on an additional parameter $n$)…

Let $n$ be a positive integer and let $C_n$ be the cycle indicator of the symmetric group $S_n$. Carlitz proved that if $p$ is a prime, and if $r$ is a non negative integer, then we have the congruence $C_{r+np}\equiv (X_1^p-X_p)^nC_r…

Hasanalizade [1] studied Deaconescu's conjecture for positive composite integer $n$. A positive composite integer $n\geq4$ is said to be a Deaconescu number if $S_2(n)\mid \phi(n)-1$. In this paper, we improve Hasanalizade's result by…

It is shown that for any prime $p$ and any natural numbers $\ell, m,$ and $s$ such that $0<s<p$, the three following congruences \begin{align*}\sum_{i\ge \ell+1}(-1)^{m-i} {m \choose i}{m+s-1+i(p-1) \choose m+s-1+\ell(p-1)} &\equiv 0 \bmod…

We prove in a direct fashion that a multidimensional probability measure is determinate if the higher dimensional analogue of Carleman's condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the…

We prove some supercongruence and divisibility results on sums involving Domb numbers, which confirm four conjectures of Z.-W. Sun and Z.-H. Sun. For instance, by using a transformation formula due to Chan and Zudilin, we show that for any…

Docovic and Szechtman, [Proc. Amer. Math. Soc. 133 (2005) 2853-2863] considered a vector space V endowed with a bilinear form. They proved that all isometries of V over a field F of characteristic not 2 have determinant 1 if and only if V…

We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…

Let $p$ be an odd prime, Jianqiang Zhao has established a curious congruence, which is $$ \sum_{i+j+k=p \atop i,j,k > 0} \frac{1}{ijk} \equiv -2B_{p-3}\pmod p , $$ where $B_{n}$ denotes the $n$-th Bernoulli number. In this paper, we will…

The Deligne-Simpson problem in the multiplicative version is formulated like this: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\in SL(n,{\bf C})$ so that there exist irreducible $(p+1)$-tuples…

Deciding the positivity of a sequence defined by a linear recurrence and initial conditions is, in general, a hard problem. When the coefficients of the recurrences are constants, decidability has only been proven up to order 5. The…

We study the validity of congruence inclusions of the form $ \alpha ( \beta \circ \alpha \gamma \circ \beta \circ \dotsc \circ \alpha \gamma \circ \beta ) \subseteq \alpha \beta \circ \alpha \gamma \circ \alpha \beta \circ \dots$ in…