The abc conjecture of Oesterlé and Masser

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. The abc-conjecture of Oesterlé and Masser

   For any given $\varepsilon>0$, let $a,b,c$ be coprime positive integers satisfying $a+b=c$, and let $R(N)$ denote the product of the distinct prime divisors of $N$. Oesterlé–Masser abc-conjecture. One has

   $$c\ll_{\varepsilon}R(abc)^{1+\varepsilon}.$$

   This conjecture is used in the paper to study the Erdős–Woods conjecture and related Diophantine questions; the source subsequently assumes an explicit form of the conjecture.

   source: Tarlok N. Shorey and Rob Tijdeman, “Arithmetic properties of blocks of consecutive integers”, arXiv:1612.05438 (2016).

2. The abc conjecture of Oesterlé and Masser

   Let $a$, $b$, and $c$ be relatively prime integers satisfying $a+b=c$, and define the radical of a positive integer $n$ by

   $$\operatorname{rad}(n)=\prod_{p\mid n}p.$$

   The abc conjecture. For every $\epsilon>0$, only finitely many triples $(a,b,c)$ fail to satisfy

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

   The paper applies this conjecture with $\epsilon=1/6$ to prove conditional finiteness results for repunit $s$-Cullen numbers. The conjecture remains open.

   source: Jon Grantham and Hester Graves, “The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits”, arXiv:2009.04052 (2021).