Walking to infinity on gaussian lines

Walking to Infinity on the Fibonacci Sequence

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 Fibonacci sequence. We prove that all walks starting with a Fibonacci number and the following terms are Fibonacci numbers obtained by appending exactly one digit at a time to the right have a length of at most two. In the more general case where we append at most a bounded number of digits each time, we give a formula for the length of the longest walk.

沿某些数论序列走向无穷

Walking to Infinity Along Some Number Theory sequences

本文研究沿逐项增加一位数字的素数等数论序列走向无穷的猜想，通过贪心与随机模型分析轨迹行为和游走长度，并证明若干序列及其他进制下不可能。

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。

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…

On the Distribution of the Greatest Common Divisor of Gaussian Integers

For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with explicit error terms. We also present results for higher moments along with computational data which support the results for the second, third, fourth, and fifth moments. The analogous question for integers is studied by Diaconis and Erdös.