# Questions tagged [abc-conjecture]

For questions about and related to the abc conjecture.

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### What is the implication of a particular choice of $k(\epsilon)$ for the ABC Conjecture.

Consider the $ABC$ conjecture in the following form: For every positive real number $\epsilon$, there exists a constant $k(\epsilon)$ such that for all triples $(a, b, c)$ of coprime positive ...
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### Can we prove that for ABC-triples the product $A*B*C$ is unique?

Consider positive coprime integers $A$ and $B$ with $A+B=C$. The triple $(A,B,C)$ is called an ABC-triple if the radical of the product $ABC$ is smaller then $C$. The radical of a positive integer $n$,...
126 views

### The $abc$ conjecture as a special case of Vojta's height inequality

From Quanta I've learned that Peter Scholze and Jakob Stix rejected Shinichi Mochizuki's proof of the $abc$ conjecture in September 2018. As a non-expert one stumbles a little earlier than necessary ...
138 views

### ABC conjecture and an inequality

Problem: Let $p,q,r$, be positive integers satisfying $\frac {1}{p} + \frac {1}{q} + \frac {1}{r} < 1$ . If the ABC conjecture is true, then $x^p + y^q = z^r$ has finitely many positive integer ...
### given $p,q,r \ge 3$ study the diophantine equation $x^py^q=z^r-1$ using the $abc$-conjecture
I want to show that given $p,q,r \ge 3$ the diophantine equation $x^py^q=z^r-1$ has only finitely many solutions with $x,y,z \in \mathbb{N} = 1 ,2, \dots$ assuming the $abc$-conjecture. The proof ...