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.

assumingthe latter as a black box!) gives horrifically gigantic constants (massive exponentials, etc.). For this and other reasons, the proof of FLT via modularity methods seems likely to remain the only way to prove FLT and to understand why it is true. $\endgroup$thatinconceivable that ABC implies FLT, even though not in the obvious way - we might e.g. be able to prove that if there is a solution with exponent $n$, then there is a solution with an exponent $m>n$, in which case we would get a contradiction with ABC. I am not claiming I know how to prove FLT from ABC this way, just sharing a thought. $\endgroup$3more comments