Related papers: Walking through the Gaussian Primes
An interesting question, known as the Gaussian moat problem, asks whether it is possible to walk to infinity on Gaussian primes with steps of bounded length. Our work examines a similar situation in the real quadratic integer ring…
The Gaussian Moat Problem asks whether it is possible to walk from the origin to infinity in the complex plane using only Gaussian primes as stepstones and steps of bounded length. We prove that this is not possible.
We generalize the Gaussian moat problem to the finitely many imaginary quadratic fields exhibiting unique factorization in their integers. We map this problem to the Euclidean minimum spanning tree problem and use Delaunay triangulation and…
The Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. In this paper, we have proved that the…
In this paper, we have developed an algorithm for the prime searching in $\mathbb{R}^3$. This problem was proposed by M. Das [Arxiv,2019]. This paper is an extension of her work. As we know the distribution of primes will get more irregular…
An interesting open conjecture asks whether it is possible to walk to infinity along primes, where each term in the sequence has one digit more than the previous. We present different greedy models for prime walks to predict the long-time…
We study analogies between the rational integers on the real line and the Gaussian integers on other lines in the complex plane. This includes a Gaussian analog of Bertrands Postulate, the Chinese Remainder Theorem, and the periodicity of…
In recent years, computer simulations are playing a fundamental role in unveiling some of the most intriguing features of prime numbers. In this work, we define an algorithm for a deterministic walk through a two-dimensional grid that we…
Gaussian processes retain the linear model either as a special case, or in the limit. We show how this relationship can be exploited when the data are at least partially linear. However from the perspective of the Bayesian posterior, the…
Given a circle of radius $r$ centered at the origin, the Gauss Circle Problem concerns counting the number of lattice points $C(r)$ within this circle. It is known that as $r$ grows large, the number of lattice points approaches $\pi r^2$,…
We consider the problem of computing the probability of maximality (PoM) of a Gaussian random vector, i.e., the probability for each dimension to be maximal. This is a key challenge in applications ranging from Bayesian optimization to…
An interesting open problem in number theory asks whether it is possible to walk to infinity on primes, where each term in the sequence has one more digit than the previous. In this paper, we study its variation where we walk on the…
The computation of Gaussian orthant probabilities has been extensively studied for low-dimensional vectors. Here, we focus on the high-dimensional case and we present a two-step procedure relying on both deterministic and stochastic…
We analyze the Gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation. We justify the use of this approximation with ideas coming from van Kampen's expansion approach…
Assuming a uniform $q$-variant of the prime $k$-tuple conjecture, we compute moments of the number of primes in arithmetic progressions to a large modulus $q$ as the residue classes vary. Consequently, depending on the size of $\varphi(q)$,…
We introduce and consider a certain probability question involving elementary number theory and the likelihood that a fixed prime will appear in a certain recursively defined factorization of an integer. We derive several convergent…
A new algorithm is developed to tackle the issue of sampling non-Gaussian model parameter posterior probability distributions that arise from solutions to Bayesian inverse problems. The algorithm aims to mitigate some of the hurdles faced…
Gaussian comparison theorems are useful tools in probability theory; they are essential ingredients in the classical proofs of many results in empirical processes and extreme value theory. More recently, they have been used extensively in…
Insight into a number of interesting questions in cosmology can be obtained from the first crossing distributions of physically motivated barriers by random walks with correlated steps. We write the first crossing distribution as a formal…
The aim of this paper is to develop novel quantum algorithms for Gaussian process quadrature methods. Gaussian process quadratures are numerical integration methods where Gaussian processes are used as functional priors for the integrands…