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 $\operatorname{rad}(abc)<c$, where $\operatorname{rad}(n)$ 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 $\operatorname{rad}(abc)^{1+\epsilon}<c$. 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.