Related papers: On the last digits of consecutive primes
Most prime gaps results have been proven using tools from analytic or algebraic number theory in the last few centuries. In this paper, we would like to present some probabilistic way of proving many essential results. A major component of…
Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as…
Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense…
We present an algorithm analogous to the sieve of Eratosthenes that produces the list of twin primes. Next, we count the number of twin primes resulting from the construction with two different heuristic arguments. The first method is…
We study the posterior contraction behavior of the latent population structure that arises in admixture models as the amount of data increases. We adopt the geometric view of admixture models - alternatively known as topic models - as a…
We investigate by means of a simple theoretical model the emergence of prime numbers as life cycles, as those seen for some species of cicadas. The cicadas, more precisely, the Magicicadas spend most of their lives below the ground and then…
In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the…
This contribution is concerned with mathematical models for the dynamics of the genetic composition of populations evolving under recombination. Recombination is the genetic mechanism by which two parent individuals create the mixed type of…
In this paper we consider a slightly different sieve method from Eratosthenes' to get primes. We find the periodicity and mirror symmetry of the pattern.
Infinite population models are important tools for studying population dynamics of evolutionary algorithms. They describe how the distributions of populations change between consecutive generations. In general, infinite population models…
We have devised an alternative approach to sifting integers in the sieve of Eratosthenes that helps refine the error term. Instead of eliminating all multiples of a prime number $p<z$ in the traditional sieve method, our approach solely…
Recently we have introduced a novel characterisation of the distribution of twin primes that consists of three essential elements. These are: that the twins are most naturally viewed as a subsequence of the primes themselves, that the…
Prime numbers appeared in contexts spanning statistical mechanics, quantum mechanics and dynamical systems. However, the mechanisms governing the irregularities observed in their sequence and linking them to physical systems remained…
We show that there exists pairs of consecutive primes less than $x$ whose difference is larger than $t(1+o(1))(\log{x})(\log\log{x})(\log\log\log\log{x})(\log\log\log{x})^{-2}$ for any fixed $t$. Our proof works by incorporating recent…
We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers…
While generic competitive systems exhibit mixtures of hierarchy and cycles, real-world systems are predominantly hierarchical. We demonstrate and extend a mechanism for hierarchy; systems with similar agents approach perfect hierarchy in…
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,…
We describe a continuous-time modelling framework for biological population dynamics that accounts for demographic noise. In the spirit of the methodology used by statistical physicists, transitions between the states of the system are…
The goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals. We model the population by a measure space where the measure determines the abundance of individual…
Prime numbers have fascinated mathematicians since antiquity, with ongoing efforts to uncover both their properties and ever-larger examples. While giant primes rarely aid cryptography, they find use in areas such as locally decodable…