Published August 25, 2023 | Version v1
Journal article Open

On abc Triples of the Form (1,c1,c)

  • 1. University of California, Santa Barbara
  • 2. University of St. Thomas
  • 3. University of Michigan
  • 4. Colorado State University

Description

By an abc triple, we mean a triple (a,b,c) of relatively prime positive integers a,b, and c such that a+b=c and rad(abc)<c, where rad(n) denotes the product of the distinct prime factors of n. The study of abc triples is motivated by the abc conjecture, which states that for each ϵ>0, there are finitely many abc triples (a,b,c) such that rad(abc)1+ϵ<c. The necessity of the ϵ in the abc conjecture is demonstrated by the existence of infinitely many abc triples. For instance, (1,9k1,9k) is an abc triple for each positive integer k. In this article, we study abc triples of the form (1,c1,c) and deduce two general results that allow us to recover existing sequences of abc triples having a=1 that are in the literature.

Files

x64.pdf

Files (416.5 kB)

Name Size Download all
md5:36b36031671078afa2d3ab301fe5758d
416.5 kB Preview Download