  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.
