Related papers: Counting Fixed Points, Two-Cycles, and Collisions …
For a prime $p$ and a positive integer $s$ consider a homogeneous linear system over the ring $\mathbb{Z}_{p^s}$ (the ring of integers modulo $p^s$) described by an $n \times m$-matrix. The possible number of solutions to such a system is…
In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some…
We provide a formula for the number of $\mathbb{F}_{p}$-points on the Dwork hypersurface $$x_1^n + x_2^n \dots + x_n^n - n \lambda \, x_1 x_2 \dots x_n=0$$ in terms of a $p$-adic hypergeometric function previously defined by the author.…
Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…
Let $P = A\times A \subset \mathbb{F}_p \times \mathbb{F}_p$, $p$ a prime. Assume that $P= A\times A$ has $n$ elements, $n<p$. See $P$ as a set of points in the plane over $\mathbb{F}_p$. We show that the pairs of points in $P$ determine…
In this article we present method of solving some additive problems with primes. The method may be employed to the Goldbach-Euler conjecture and the twin primes conjecture. The presented method also makes it possible to obtain some…
Heegner cycles are higher weight analogues of Heegner points. Their arithmetic intersection numbers also appear as Fourier coefficients of modular forms and often belong to abelian extensions of imaginary-quadratic fields. Rotger and Seveso…
The discrete logarithm is a problem that surfaces frequently in the field of cryptography as a result of using the transformation g^a mod n. This paper focuses on a prime modulus, p, for which it is shown that the basic structure of the…
We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by…
Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also study…
We have shown previously that at each stage of Eratosthenes sieve there is a corresponding cycle of gaps $\mathcal{G}(p_0^\#)$. We can view these cycles of gaps as a discrete dynamic system, and from this system we can obtain exact models…
Let \(u\neq \pm 1,v^2\) be a fixed integer, let \(p\geq 2\) be a prime, and let $\text{ord}_p(u) \mid p-1$ be the multiplicative order of $u \text{ mod } p$. Define a prime counting function by $\pi(u,x)=\# \{ p\leq x:\text{ord}_p(u)=p-1…
The Welch map $x \rightarrow g^{x-1+c}$ is similar to the discrete exponential map $x \rightarrow g^x$, which is used in many cryptographic applications including the ElGamal signature scheme. This paper analyzes the number of solutions to…
$\beta(1,0)$-trees provide a convenient description of rooted non-separable planar maps. The involution $h$ on $\beta(1,0)$-trees was introduced to prove a complicated equidistribution result on a class of pattern-avoiding permutations. In…
We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central…
This paper introduces a methodology to derive explicit power series approximations for the limit cycle periodic solutions of the Hopf bifurcation in autonomous discrete delay differential equations (DDE). The procedure extends the…
We prove a couple of related theorems including Legendre's and Andrica's conjecture. Key to the proofs is an algorithm that delivers the exact upper bound on the greatest gap that can occur in a combinatorial game with the set of P primes…
An augmented generalized happy function $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. In this paper, we study various properties of the fixed points of $S_{[c,b]}$; count the number of fixed…
A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. In order to count the decomposables, one wants to know, under a suitable normalization, the number of equal-degree collisions of…
This work proposes elementary proofs of several related primes counting problems, based on an elementary weighted sieve. The subsets of primes considered here are the followings: the subset of twin primes PT = {p and p + 2 are primes}, the…