Let $\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of $A$ ? (2) When can one represent every element of $\Z/pZ$ as a sum of some elements of $A$ ? (3) When can one represent every element of $\Z/pZ$ as a sum of $l$ elements of $A$ ?
In this paper we start to investigate a new body of questions in additive combinatorics. The fundamental Cauchy--Davenport theorem gives a lower bound on the size of a sumset A+B for subsets of the cyclic group Zp of order p (p prime), and this is just one example of a large family of results. Our aim in this paper is to investigate what happens if we restrict the number of elements of one set that we may use to form the sums. Here is the question we set out to answer: given two subsets, A and B, does B have a subset C of bounded size such that A+C is large, perhaps even comparable to the size…
The classical Cauchy--Davenport inequality gives a lower bound for the size of the sum of two subsets of ${\mathbb Z}_p$, where $p$ is a prime. Our main aim in this paper is to prove a considerable strengthening of this inequality, where we take only a small number of points from each of the two subsets when forming the sum. One of our results is that there is an absolute constant $c>0$ such that if $A$ and $B$ are subsets of ${\mathbb Z}_p$ with $|A|=|B|=n\le p/3$ then there are subsets $A'\subset A$ and $B'\subset B$ with $|A'|=|B'|\le c \sqrt{n}$ such that $|A'+B'|\ge 2n-1$. In fact, we sho…
We show that, modulo some odd prime p, the powers of two weighted sum of the first p-2 divided Bernoulli numbers equals the Agoh-Giuga quotient plus twice the number of permutations on p-2 letters with an even number of ascents and distinct from the identity. We provide a combinatorial characterization of Wieferich primes, as well as of primes p for which p^2 divides the Fermat quotient q_p(2).
Congruences of Multipartition Functions Modulo Powers of Primes
令$p_r(n)$表示$n$的$r$ -分量多重划分的数目，并且令$S_{γ,λ}$是由$η(24z)^γϕ(24z)$张成的空间，其中$η(z)$是Dedekind的eta函数，并且$ϕ(z)$是$M_λ({\rm SL}_2(\mathbb{Z}))$中的全纯模形式。本文证明了$p_r(\frac{m^k n +r}{24})$关于$n$的生成函数与空间$S_{γ,λ}$模$m^k$中的一个函数全等.作为特殊情况，这种关系导致许多众所周知的同余，包括$p(n)$模$5,7,11$的拉马努金同余和$p_2(n)$模5和$p_{8}(n)$模11的甘地同余。进一步，利用$S_{γ,λ}$在Hecke算子$T_{\ell^2}$下的不变性，得到了与$p_r(n)$的$m^k$-adic性质有关的两类同余.
