Extending a problem of Pillai to Gaussian lines

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.