On $abc$ triples of the form $(1,c-1,c)$

Abstract

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 $\mathrm{rad}(abc)<c$, where $\mathrm{rad}(n)$ 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 $\epsilon >0$, there are finitely many $abc$ triples $(a,b,c)$ such that $\mathrm{rad}(abc{)}^{1+\epsilon }<c$. The necessity of the $\epsilon$ in the $abc$ conjecture is demonstrated by the existence of infinitely many $abc$ triples. For instance, $\left(1,9^k-1,9^k\right)$ is an $abc$ triple for each positive integer $k$. In this article, we study $abc$ triples of the form $\left(1,c-1,c\right)$ and deduce two general results that allow us to recover existing sequences of $abc$ triples having $a=1$ that are in the literature.

arXiv comment18 pages; incorporates referee's suggestions; a small correction to the statement of Lemma 2.7; final version to appear in Integers