关于高斯护城河问题的一个注记

关于高斯护城河问题

On the Gaussian Moat Problem

本文证明高斯护城河问题的答案是否定的，即无法仅以高斯素数为踏脚石、以有界步长从原点走向无穷远。

论高斯护城河问题的推广

On the Extension of the Gaussian Moat Problem

本文提出在三维空间中搜索素数的算法，推广了 M. Das 的工作，并论证了高斯护城河问题无法推广到四维以上空间的原因。

穿越高斯素数

Walking through the Gaussian Primes

本文研究高斯护城河问题，提出在平面上高效寻找高斯素数的算法，计算护城河并分析时间复杂度，与既有算法比较，同时将素性概念推广到三维空间并提出新问题。

Walks on Primes in Imaginary Quadratic Fields

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 Kruskal's algorithm to find moats in these fields in $O(|\mathbf{P}| \log |\mathbf{P}|)$ time, where $\mathbf{P}$ denotes the set of primes we are searching through. Our algorithm uses symmetry and boundaries to correctly and efficiently identify all moats, in order of increasing length, along with the farthest prime reached for each moat, in an underlying …

Modeling Random Walks to Infinity on Primes in $\mathbb{Z}[\sqrt{2}]$

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 $\mathbb{Z}[\sqrt{2}]$ whose primes cluster near the asymptotes $y = \pm x/\sqrt{2}$ as compared to Gaussian primes, which cluster near the origin. We construct a probabilistic model of primes in $\mathbb{Z}[\sqrt{2}]$ by applying the prime number theorem and a combinatorial theorem for counting the number of lattice points whose absolute values of their norms are at…

A note on Diophantine approximation with Gaussian primes

We investigate the distribution of $pθ$ modulo 1, where $θ$ is a complex number which is not contained in $\mathbb{Q}(i)$, and $p$ runs over the Gaussian primes.