Related papers: Explicit inequalities for the nth lucky number

We create a simple test for distinguishing between sets of primes and random numbers using just the sum-of-digits function. We find that the sum-of-the-digits of prime numbers does not have an equal probability of being odd or even. The…

Let u(n) be the n-th term of a Lucas sequence or a Lehmer sequence.In this article we shall establish an estimate from below for the greatest prime factor of u(n) which is of the form nexp(logn/104loglogn). In so doing we are able to…

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences $(u_n)_{n\geq 0}$ in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we…

Our main source of inspiration was a talk by Hendrik Lenstra on harmonic numbers, which are numbers whose only prime factors are two or three. Gersonides proved 675 years ago that one can be written as a difference of harmonic numbers in…

We study the distribution of primes from a topological viewpoint. Certain conjecture is introduced, and we show that it is equivalent to the Riemann Hypothesis.

For an integer $k\geq 2$, let $(L_{n}^{(k)})_{n}$ be the $k-$generalized Lucas sequence which starts with $0,\ldots,0,2,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. In this paper, we find all the integers…

The law of large numbers (LLN) and central limit theorem (CLT) are long and widely been known as two fundamental results in probability theory. Recently problems of model uncertainties in statistics, measures of risk and superhedging in…

The distribution of lattice points with relatively $r$-prime is related to problems in the Number Theory such as the Extended Lindel\"{o}f Hypothesis and the Gauss Circle Problem. It is known that Sittinger's result is improved on the…

This document presents an alternative proof of Sylvester's theorem stating that "the product of $n$ consecutive numbers strictly greater than $n$ is divisible by a prime strictly greater than $n$". In addition, the paper proposes stronger…

We solve some famous conjectures on the distribution of primes. These conjectures are to be listed as Legendre's, Andrica's, Oppermann's, Brocard's, Cram\'{e}r's, Shanks', and five Smarandache's conjectures. We make use of both…

Let $q$ be an odd prime power and $n$ be a positive integer. Let $\ell\in \mathbb F_{q^n}[x]$ be a $q$-linearised $t$-scattered polynomial of linearized degree $r$. Let $d=\max\{t,r\}$ be an odd prime number. In this paper we show that…

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira on the asymptotic distribution of primitive rational points on expanding closed horospheres in the space of lattices. Key ingredients of our proof include…

The appearance of primes in a family of linear recurrence sequences labelled by a positive integer $n$ is considered. The terms of each sequence correspond to a particular class of Lehmer numbers, or (viewing them as polynomials in $n$)…

The well-known sequence $\vartheta_n = \vartheta(p_n) = \sum_{i=1}^n \ln p_i= \ln\left([p_n]\#\right)$ exhibits numerous extremely interesting properties. Since $p_n = \exp(\vartheta_n - \vartheta_{n-1})$, it is immediately clear that the…

We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [Electronic Journal of Combinatorics…

Let $k\ge 1$ be an integer. A positive integer $n$ is $k$-\textit{gleeful} if $n$ can be represented as the sum of $k$th powers of consecutive primes. For example, $35=2^3+3^3$ is a $3$-gleeful number, and $195=5^2+7^2+11^2$ is $2$-gleeful.…

The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $\pi(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at…

Using a sieve-theoretic argument, we show that almost all gaps $(p_n, p_{n+1})$ between consecutive primes $p_n, p_{n+1}$ contain a natural number $m$ whose least prime factor $p(m)$ is at least the length $p_{n+1} - p_n$ of the gap,…

A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences…

In this article we take a probabilistic look at H\"older's inequality, considering the ratio of terms in the classical H\"older inequality for random vectors in $\mathbb{R}^n$. We prove a central limit theorem for this ratio, which then…