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Congruences of Multipartition Functions Modulo Powers of Primes
modular form partition multipartition Ramanujan-type congruence
2014/6/3
Let pr(n) denote the number of r-component multipartitions of n, and let Sγ λ be the space spanned by η(24z)γφ(24z), where η(z) is the Dedekind’s eta function and η(z) is a holomorphic modular form i...
New properties of multiple harmonic sums modulo $p$ and $p$-analogues of Leshchiner's series
Congruence finite central binomial sum multiple harmonic sum Bernoulli number
2012/6/15
In this paper we present some new identities of hypergeometric type for multiple harmonic sums whose indices are the sequences $(\{1\}^a,c,\{1\}^b),$ $(\{2\}^a,c,\{2\}^b)$ and prove a number of congru...
Periods of rational maps modulo primes
p-adic dynamics Mordell-Lang conjecture Algebraic Geometry
2011/9/6
Abstract: Let $K$ be a number field, let $\phi \in K(t)$ be a rational map of degree at least 2, and let $\alpha, \beta \in K$. We show that if $\alpha$ is not in the forward orbit of $\beta$, then th...
Homological algebra modulo exact zero-divisors
Exact zero divisors (co)homology complexity
2011/1/21
We study the homological behavior of modules over local rings modulo exact zero-divisors. We obtain new results which are in some sense “opposite” to those known for modules over local rings modulo re...
This article proves a conjecture by S.-C. Liu and C.-C. Yeh about Catalan numbers, which states that odd Catalan numbers can take exactly k − 1 distinct values modulo 2k, namely the values C21 &...
An Iterative Construction of Irreducible Polynomials Reducible Modulo Every Prime
Iterative Construction Irreducible Polynomials Reducible Modulo Every Prime
2011/1/21
We give a method of constructing polynomials of arbitrarily large degree irreducible over a global field F but reducible modulo every finite prime of F.The method consists of finding quadratic f ∈ F[x...
Universal Communication over Modulo-additive Channels with an Individual Noise Sequence
Universal Communication Modulo-additive Channels Individual Noise Sequence
2011/1/21
Which communication rates can be attained over an unknown channel where the relation between the input and output can be arbitrary? A channel where the output is any arbitrary (possibly stochastic) fu...
Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2
the Erdos-Moser equation math
2010/11/15
An open conjecture of Erdos and Moser is that the only solution of the Diophantine equation in the title is the trivial solution 1+2=3. Reducing the equation modulo k and k^2, we give necessary and s...
Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences
Andrews' spt-function Atkin's Hecke-type partition congruences
2010/11/15
New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of ...
Congruences for Andrews' spt-function modulo powers of 5, 7 and 13
Andrews' spt-function math
2010/11/15
Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 a...
Some new super-congruences modulo prime powers
super-congruences modulo prime powers math
2010/11/22
Let $p>3$ be a prime. We show that $$\sum_{k=0}^{p-1}\binom(-1/(p+1),k)^{p+1}=0 (mod p^5).$$ For any positive integer m not divisible by p, we prove that $$\sum_{k=0}^{p-1}(-1)^{km}\binom(p/m-1,k)^m=0...
Level lowering modulo prime powers and twisted Fermat equations
Modular forms level lowering Diophantine equations
2010/11/26
We discuss a clean level lowering theorem modulo prime powers for weight 2 cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations axn + byn + ...
We show that for infinitely many prime numbers p there are at least \log\log p / \log\log\log p distinct residue classes modulo p that are not congruent to n! for any integer n.
Set Systems with L-intersections modulo a Prime Number
Erdö s-Ko-Rado Theorem Frankl-Ray-Chaudhuri-Wilson Theorems Frankl-Füredi’s conjecture Snevily's conjecture
2014/6/3
Let p be a prime and let L = {l1, l2,..., ls} and K = {k1, k2,... kr} be two subsets of {0, 1, 2,..., p-1} satisfying max lj < min ki. We will prove the following results: If F = {F1, F2,..., Fm} is a...
An Optical Interferometer for Parallel Modulo Operation
Prime factorization Optical parallel processing Interference
2009/6/2
An optical processor for modulo operations is estimated. Large scale information processing for modulo operations is effective in an algorithm in prime factorization. This processor is based on a meth...