Agoh's conjecture: its generalizations, its analogues

In this paper we formulate two generalizations of Agoh's conjecture. We also formulate conjectures involving congruence modulo primes about hyperbolic secant, hyperbolic tangent, N\"orlund numbers, as well as about coefficients of expansions in powers of other analytic functions. We formulate a thesis about combinatorial objects that do not produce fake primes.Comment: 9 page

Congruences involving binomial coefficients and Ap\'ery-like numbers

For $n=0,1,2,\ldots$ let $W_n=\sum_{k=0}^{[n/3]}\binom{2k}k \binom{3k}k\binom n{3k}(-3)^{n-3k}$, where $[x]$ is the greatest integer not exceeding $x$. Then $\{W_n\}$ is an Ap\'ery-like sequence. In this paper we deduce many congruences involving $\{W_n\}$, in particular we determine $\sum_{k=0}^{p-1}\binom{2k}k\frac{W_k}{m^k}\pmod p$ for $m=-640332,-5292,-972,-108,-44,-27,-12,8,54,243$ by using binary quadratic forms, where $p>3$ is a prime. We also prove several congruences for generalized Ap\'ery-like numbers, and pose 29 challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.Comment: 28 page

Conjectures on representations involving primes

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists $k\in\{0,\ldots,n\}$ such that $n+k$ and $n+k^2$ are both prime. (ii) Each integer $n>1$ can be written as $x+y$ with $x,y\in\{1,2,3,\ldots\}$ such that $x+ny$ and $x^2+ny^2$ are both prime. (iii) For any rational number $r>0$, there are distinct primes $q_1,\ldots,q_k$ with $r=\sum_{j=1}^k1/(q_j-1)$. (iv) Every $n=4,5,\ldots$ can be written as $p+q$, where $p$ is a prime with $p-1$ and $p+1$ both practical, and $q$ is either prime or practical. (v) Any positive rational number can be written as $m/n$, where $m$ and $n$ are positive integers with $p_m+p_n$ a square (or $\pi(m)\pi(n)$ a positive square), $p_k$ is the $k$-th prime and $\pi(x)$ is the prime-counting function.Comment: 33 pages, final published versio

Problems on combinatorial properties of primes

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find that $\pi(x)$ and $p_n$ have many combinatorial properties which should not be ignored. In this paper we pose 60 open problems on combinatorial properties of primes (including connections between primes and partition functions) for further research. For example, we conjecture that for any integer $n>1$ one of the $n$ numbers $\pi(n),\pi(2n),...,\pi(n^2)$ is prime; we also conjecture that for any integer $n>6$ there exists a prime $p<n$ such that $pn$ is a primitive root modulo $p_n$. One of our conjectures involving the partition function $p(n)$ states that for any prime $p$ there is a primitive root $g<p$ modulo $p$ with $g\in\{p(n):\ n=1,2,3,...\}$.Comment: 19 pages. Correct the typo 2k+1 in Conj. 3.21(i) as 2k-1. In: Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), World Sci., Singapore, 2015, pp. 169--18

Leading Digits of Mersenne Numbers

It has long been known that sequences such as the powers of $2$ and the factorials satisfy Benford's Law; that is, leading digits in these sequences occur with frequencies given by $P(d)=\log_{10}(1+1/d)$, $d=1,2,\dots,9$. In this paper, we consider the leading digits of the Mersenne numbers $M_n=2^{p_n}-1$, where $p_n$ is the $n$-th prime. In light of known irregularities in the distribution of primes, one might expect that the leading digit sequence of $\{M_n\}$ has \emph{worse} distribution properties than "smooth" sequences with similar rates of growth, such as $\{2^{n\log n}\}$. Surprisingly, the opposite seems to be the true; indeed, we present data, based on the first billion terms of the sequence $\{M_n\}$, showing that leading digits of Mersenne numbers behave in many respects \emph{more regularly} than those in the above smooth sequences. We state several conjectures to this effect, and we provide an heuristic explanation for the observed phenomena based on classic models for the distribution of primes.Comment: 23 pages, 29 figure

Congruences for two types of Apery-like sequences

In this paper we present many results and conjectures on congruences involving two types of Ap\'ery-like sequences $\{G_n(x)\}$ and $\{V_n(x)\}$.Comment: 47 page

Congruences for Domb and Almkvist-Zudilin numbers

In this paper we prove some transformation formulae for congruences modulo a prime and deduce some congruences for Domb numbers and Almkvist-Zudilin numbers. We also pose some conjectures on congruences modulo prime powers.Comment: 17 page

An explicit theory of π1un,crys(P1−{0,μN,∞})\pi_{1}^\mathrm{un,crys}(\mathbb{P}^{1} - \{0,\mu_{N},\infty\}) - II-3 : Sequences of multiple harmonic sums viewed as periods

Let $X=\text{ }\mathbb{P}^{1} - (\{0,\infty\} \cup \mu_{N})\text{ }/\text{ }W(\mathbb{F}_{q})$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p$ prime to $N$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline Frobenius of the pro-unipotent fundamental groupoid of $X$. In part I, we have computed explicitly the Frobenius action. In part II, we use this computation to understand explicitly the algebraic relations of cyclotomic $p$-adic multiple zeta values. We have used the ideas and the vocabulary of the Galois theory of periods, and in our framework, certain sequences of prime weighted multiple harmonic sums have been dealt with as if they were periods. In this II-3, we define three notions which essentialize our three types of computations, respectively : a "continuous" groupoid \pi_{1}^{\un,\DR}(X_{K})^{\hat{\text{cont}}}, a "localization" \pi_{1}^{\un,\DR}(X_{K})^{\loc} of \pi_{1}^{\un,\DR}(X_{K}), and a "rational counterpart at zero" \pi_{1}^{\un,\RT,0}(X_{K}) of \pi_{1}^{\un,\DR}(X_{K}). As an application, and as a conlcusion of this part II, we justify and clarify our Galois-theoretic point of view, in particular, we construct period maps and state period conjectures for sequences of prime weighted multiple harmonic sums.Comment: 43 page

Supercongruences involving dual sequences

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom nk(-1)^k\binom xk\binom{-1-x}k.$$ For any odd prime $p$ and $p$-adic integer $x$, we determine $\sum_{k=0}^{p-1}(\pm1)^kd_k(x)^2$ and $\sum_{k=0}^{p-1}(2k+1)d_k(x)^2$ modulo $p^2$; for example, we establish the new $p$-adic congruence $$\sum_{k=0}^{p-1}(-1)^kd_k(x)^2\equiv(-1)^{\langle x\rangle_p}\pmod{p^2},$$ where $\langle x\rangle_p$ denotes the least nonnegative integer $r$ with $x\equiv r\pmod p$. For any prime $p>3$ and $p$-adic integer $x$, we determine $\sum_{k=0}^{p-1}s_k(x)^2$ modulo $p^2$ (or $p^3$ if $x\in\{0,\ldots,p-1\}$), and show that $$\sum_{k=0}^{p-1}(2k+1)s_k(x)^2\equiv0\pmod{p^2}.$$ We also pose several related conjectures.Comment: 35 pages, final published versio

Curious congruences for Fibonacci numbers

In this paper we establish some sophisticated congruences involving central binomial coefficients and Fibonacci numbers. For example, we show that if $p\not=2,5$ is a prime then $$\sum_{k=0}^{p-1}F_{2k}\binom{2k}{k}=(-1)^{[p/5]}(1-(p/5)) (mod p^2)$$ and $$\sum_{k=0}^{p-1}F_{2k+1}\binom{2k}k=(-1)^{[p/5]}(p/5) (mod p^2).$$ We also obtain similar results for some other second-order recurrences and raise several conjectures.Comment: 16 pages. Revise Conj. 4.