I have frequently read and heard that given the ABC-conjecture a number of important unsolved problems of number theory can be solved (with relatively simple proofs). Among them, the celebrated Fermat's Last theorem is frequently mentioned.
So, my question is: Given that the $ABC$ conjecture is valid, can we prove that it implies Fermat's Last theorem ?
P.S.: I can understand that ABC conjecture "easily" implies the asymptotic FLT (stating that: "the equ-ation $x^n+y^n=z^n$ can have solutions in positive integers only for $n< n_0$, where $n_0$ is some finite number"). This is outlined in Lang's Algebra (p.196, 1994 edition), see also here and here.
Best Answer
No, abc doesn't imply FLT.
For all exponents $n > 3$, abc implies at most finitely many counterexamples to FLT, but it allows counterexamples to FLT.
For exponent $n=3$ it allows infinitely many counterexamples and the Fermat-like equation $x^3+y^3=a z^3$ has infinitely many coprime solutions for some $a$ via the group law on the elliptic curve.
Basically finite number of abc triples of arbitrary quality don't contradict abc, while a single counterexample contradicts FLT.
Also, there is generalization of abc over number fields, widely believed to be true.
Over number fields FLT fails, e.g. $1^n+1^n=(\sqrt[n]{2})^n$.