New Congruences on Multiple Harmonic Sums and Bernoulli Numbers

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p \ge 11$ and integer $r\ge 2$, we prove that $$\sum\limits_{\begin{smallmatrix} {{l}_{1}}+{{l}_{2}}+\cdots +{{l}_{6}}={{p}^{r}} {{l}_{1}},\cdots ,{{l}_{6}}\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{1}{{{l}_{1}}{{l}_{2}}{{l}_{3}}{{l}_{4}}{{l}_{5}}{l}_{6}}}\equiv - \frac{{5!}}{18}p^{r-1}B_{p-3}^{2} \pmod{{{p}^{r}}}.$$ This extends a family of curious congruences. We also obtain other interesting congruences involving multiple harmonic sums and Bernoulli numbers.Comment: 14 pages. This version simplifies the previous version. Moreover, two important theorems and a conjecture were adde

A Curious Congruence Involving Alternating Harmonic Sums

Let $p$ be a prime and ${\mathcal{P}_{p}}$ the set of positive integers which are prime to $p$. We establish the following interesting congruence $$\sum\limits_{\begin{smallmatrix} i+j+k={{p}^{r}} i,j,k\in {\mathcal{P}_{p}} \end{smallmatrix}}{\frac{{{(-1)}^{i}}}{ijk}}\equiv \frac{{{p}^{r-1}}}{2}{{B}_{p-3}}\, (\bmod \, {{p}^{r}}).$$Comment: This is an original research article about congruences. It has submitted for publication in June 201

Congruences for sporadic sequences and modular forms for non-congruence subgroups

In the course of the proof of the irrationality of zeta(2) R. Apery introduced numbers b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}. Stienstra and Beukers showed that for the prime p > 3 Apery numbers satisfy congruence b((p-1)/2) = 4a^2-2p mod p, if p = a^2+b^2 (where a is odd). Later, Zagier found some generalizations of Apery numbers, so called sporadic sequences, and recently Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true) and conjectured the congruence for the sixth sequence. In this paper we prove that remaining congruence by studying Atkin and Swinnerton-Dyer congruences between Fourier coefficients of certain cusp form for non-congurence subgroup.Comment: 9 page

Solving congruence equations using Bernstein forms

We present a subdivision method to solve systems of congruence equations. This method is inspired in a subdivision method, based on Bernstein forms, to solve systems of polynomial inequalities in several variables and arbitrary degrees. The proposed method is exponential in the number of variables

Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2

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 sufficient conditions on solutions to the resulting congruence and supercongruence. A corollary is a new proof of Moser's result that the conjecture is true for odd exponents n. We also connect solutions k of the congruence to primary pseudoperfect numbers and to a result of Zagier. The proofs use divisibility properties of power sums as well as Lerch's relation between Fermat and Wilson quotients.Comment: 10 pages, 2 tables, submitted for publicatio

Atkin and Swinnerton-Dyer congruences and noncongruence modular forms

Atkin and Swinnerton-Dyer congruences are special congruence recursions satisfied by coefficients of noncongruence modular forms. These are in some sense $p$-adic analogues of Hecke recursion satisfied by classic Hecke eigenforms. They actually appeared in different context and sometimes can be obtained using the theory of formal groups. In this survey paper, we introduce the Atkin and Swinnerton-Dyer congruences, and discuss some recent progress on this topic

Permutation Polynomials modulo m

This paper mainly studies problems about so called "permutation polynomials modulo $m$", polynomials with integer coefficients that can induce bijections over Z_m={0,...,m-1}. The necessary and sufficient conditions of permutation polynomials are given, and the number of all permutation polynomials of given degree and the number induced bijections are estimated. A method is proposed to determine all equivalent polynomials from the induced polynomial function, which can be used to determine all equivalent polynomials that induce a given bijection. A few problems have not been solved yet in this paper and left for open study. Note: After finishing the first draft, we noticed that some results obtained in this paper can be proved in other ways (see Remark 2). In this case, this work gives different and independent proofs of related results.Comment: 21 pages, with 2 open problem

Ap\'ery-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators

We derive an expression for the value $\zeta_Q(3)$ of the spectral zeta function $\zeta_Q(s)$ studied by Ichinose and Wakayama for the non-commutative harmonic oscillator defined in the work of Parmeggiani and Wakayama using a Gaussian hypergeometric function. In this study, two sequences of rational numbers, denoted $J_2(n)$ and $J_3(n)$, which can be regarded as analogues of the Ap\'ery numbers, naturally arise and play a key role in obtaining the expressions for the values $\zeta_Q(2)$ and $\zeta_Q(3)$. We also show that the numbers $J_2(n)$ and $J_3(n)$ have congruence relations like those of the Ap\'ery numbers.Comment: 18 page

Counting roots of truncated hypergeometric series over finite fields

We consider natural polynomial truncations of hypergeometric power series defined over finite fields. For these truncations, we establish asymptotic upper bounds of order $O(p^{11/12})$ on the number of roots in the prime field $\mathbb{F}_p$. We discuss the correspondence to families of elliptic curves and K3 surfaces of certain such hypergeometric polynomials, for which sharp bounds are obtained in some cases. We include some computations to illustrate and supplement our results.Comment: Errors in paper uncorrected. Kenneth Ward passed away June 201

Large gaps between consecutive prime numbers containing square-free numbers and perfect powers of prime numbers

We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers