The abc conjecture and the existence of finite number of solutions for the Beal's problem In this page, I am not interested on the Beal conjecture itself. But I am interested on the following problem: > The author claimed: > > **Theabc conjecture would imply that there are at most finitely many counterexamples to Beal's conjecture.** My **question** is: Is there are a proof for this claim.

In fact, for any positive $\epsilon < 1/12$ the inequality given in the $abc$-conjecture $$ rad(abc) > c_{\epsilon}c^{1-\epsilon} $$ with an explicit value of $c_{\epsilon}>0$ yields an explicit upper bound on $x^p, y^q, z^r$ in any counterexample to Beal's conjecture. By the $abc$-conjecture, there are only finitely many such $(a,b,c)$-triples.