论文标题
在$ \ mathbb {z} [\ sqrt {2}] $中，建模随机步行到无限的素数。
Modeling Random Walks to Infinity on Primes in $\mathbb{Z}[\sqrt{2}]$
论文作者
论文摘要
一个有趣的问题，称为高斯护城河问题，询问是否有可能步行到无限长度的高斯素数上。我们的工作研究了真正的二次整数环$ \ mathbb {z} [\ sqrt {2}] $的类似情况，其在渐近线附近的素数集群$ y = \ y = \ pm x/\ sqrt {2} $，与高斯素质相比，在哪个群集中，始于来源。我们在$ \ mathbb {z} [\ sqrt {2}] $中构建概率模型，通过应用质量编号定理和一个组合定理来计算其规范最多是$ r^2 $的晶格值的数量。然后，我们证明，如果步行保持在距离渐近线的一定距离之内，就不可能步行到无限。最后，我们执行一些护城河计算，以表明最长的步行可能会保持在渐近线附近。因此，我们猜测，在$ \ mathbb {z} [\ sqrt {2}] $限制长度的步骤上，没有步行到无穷大。
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 most $r^2$. We then prove that it is impossible to walk to infinity if the walk remains within some bounded distance from the asymptotes. Lastly, we perform a few moat calculations to show that the longest walk is likely to stay close to the asymptotes; hence, we conjecture that there is no walk to infinity on $\mathbb{Z}[\sqrt{2}]$ primes with steps of bounded length.
