Extending a problem of Pillai to Gaussian lines

TL;DR

This paper extends Pillai's problem to Gaussian integers, proving the existence of infinitely many sequences of consecutive Gaussian integers with specific coprimality properties on primitive Gaussian lines, and analyzing their minimal lengths.

Contribution

It introduces new bounds and conditions for sequences of Gaussian integers with particular coprimality properties on Gaussian lines, extending classical number theory results.

Findings

Existence of a threshold G_L for infinite sequences with the property.

Minimal sequence length g_L is at least 7 for all lines.

Characterization of lines with g_L=7 and lines with g_L≥260,000.

Abstract

Let $L$ be a primitive Gaussian line, that is, a line in the complex plane that contains two, and hence infinitely many, coprime Gaussian integers. We prove that there exists an integer $G_L$ such that for every integer $n\geq G_L$ there are infinitely many sequences of $n$ consecutive Gaussian integers on $L$ with the property that none of the Gaussian integers in the sequence is coprime to all the others. We also investigate the smallest integer $g_L$ such that $L$ contains a sequence of $g_L$ consecutive Gaussian integers with this property. We show that $g_L\neq G_L$ in general. Also, $g_L\geq 7$ for every Gaussian line $L$, and we give necessary and sufficient conditions for $g_L=7$ and describe infinitely many Gaussian lines with $g_L\geq 260,000$. We conjecture that both $g_L$ and $G_L$ can be arbitrarily large. Our results extend a well-known problem of Pillai from the rational integers to the Gaussian integers.