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 ≥ 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 6 = G L in general. Also, g L ≥ 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 ≥ 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.