Related papers: An extension of Boyd's $p$-adic algorithm for the …

Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin…

Let \(d_k(p)\) denote the natural density of positive integers whose \(k\)-th smallest prime divisor is \(p\). Erd\H{o}s asked whether, for each fixed \(k\), the sequence \(p\mapsto d_k(p)\) is unimodal as \(p\) ranges over the primes.…

We propose a new probabilistic algorithm of $p$-adic linear regression for random sampling with digitwise noise. This includes a new probabilistic algorithm of modulo $p$ linear regression.

Estimating the number of principal components is one of the fundamental problems in many scientific fields such as signal processing (or the spiked covariance model). In this paper, we first demonstrate that, for fixed $p$, any penalty term…

In this paper we present and expand upon procedures for obtaining large d digit prime number to an arbitrary probability. We use a layered approach. The first step is to limit the pool of random number to exclude numbers that are obviously…

The study of the well-known partition function $p(n)$ counting the number of solutions to $n = a_{1} + \dots + a_{\ell}$ with integers $1 \leq a_{1} \leq \dots \leq a_{\ell}$ has a long history in combinatorics. In this paper, we study a…

Using complex methods combined with Baire's Theorem we show that one-sided extendability, extendability and real analyticity are rare phenomena on various spaces of functions in the topological sense. These considerations led us to…

Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$;…

In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi(m)/(k+1)$ residue classes in the…

This paper continues the author's previous studies on continued fractions and Heron's algorithm, as from his former JMM2017 presentation (see \cite{CF.HA}).\par\medskip Extending the notion of continued fraction to the $p$-adic fields, one…

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…

The search for values of $p$ for which the Harmonic numbers $H_{\lfloor p/6 \rfloor}$ vanish mod $p$, carried to $p < 600,000$ by Schwindt in 1983, is extended here to $p < 149,250,000,000,000$, and two new solutions are reported. (These…

In recent years, the congruence $$ \sum_{\substack{i+j+k=p\\ i,j,k>0}} \frac1{ijk} \equiv -2 B_{p-3} \pmod{p}, $$ first discovered by the last author have been generalized by either increasing the number of indices and considering the…

It is shown that the first $n$ prime numbers $p_1,...,p_n$ determine the next one by the recursion equation $$ p_{n+1} =\lim\limits_{s\to +\infty} [\prod\limits^n_{k=1} (1-\frac{1}{p^s_k}) \sum\limits^\infty_{j=1} \frac{1}{j^s} -1]^{-1/s}.…

Let $\mathbb{F}_p$ be the field of residue classes modulo a prime number $p$ and let $A$ be a nonempty subset of $\mathbb{F}_p$. In this paper we show that if $|A|\preceq p^{0.5}$, then \[ \max\{|A\pm A|,|AA|\}\succeq|A|^{13/12};\] if…

Let $p$ be a prime. We prove bounds on short Dirichlet character sums evaluated at a class of homogeneous polynomials in arbitrary dimensions. In every dimension, this bound is nontrivial for sums over boxes with side lengths as short as…

The harmonic number $H_k=\sum_{j=1}^k1/j(k=1,2,3\cdots)$ play an important role in mathematics. Let $p>3$ be a prime. In this paper, we establish a number of congruences with the form $\sum_{k=1}^{p-1}k^mH_k^n(\mod p^2)$ for…

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform…

We consider the summability of one- and multi-dimensional trigonometric Fourier series. The Fej{\'e}r and Riesz summability methods are investigated in detail. Different types of summation and convergence are considered. We will prove that…

For an $n$-bit positive integer $a$ written in binary as $$ a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$ \overleftarrow{a} =…