Extending a problem of Pillai to Gaussian lines

2022/06/30 by Elsa Magness, Brian Nugent, Magness, Elsa +3
Computer Science · Engineering · Mathematics · #11R11 #Advanced Numerical Analysis Techniques #Digital Image Processing Techniques #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2207.00045

openalex publication_date 2022/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≠ 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.

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