# Which long-standing open problems follow immediately from the proof of the abc conjecture

Some blogs/articles claim that many problems will follow immediately from the ABC theorem if it is indeed a theorem. For example Fermat's last theorem will follow in a one-page proof from the abc theorem.

My question is: can someone use the abc theorem to outline some proofs of open problems, or solved problems with difficult proofs, that are solved immediately using the abc theorem? It would be best if there is one solved proof per answer.

I am also especially interested in seeing a one-page proof of Fermat's last theorem using the ABC theorem!

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You might start by reading en.wikipedia.org/wiki/Abc_conjecture#Some_consequences –  j.c. Sep 12 '12 at 8:49
Another site: math.unicaen.fr/~nitaj/abc.html#Consequences –  joro Sep 12 '12 at 9:18
Assuming ABC, one can prove that there are infinitely many primes $p$ such that $2^{p-1}\not\equiv1\pmod{p^2}$. The proof isn't particularly hard, but not completely trivial. Of course, we expect most primes to have this property, but even the weaker statement has eluded unconditional proof. The fact that ABC implies this is in J. Number Theory 30 (1988), 226-237. –  Joe Silverman Sep 12 '12 at 13:28
A general reference that is perhaps of interest to get a flavor of variations of ABC and equivalent formulations, a proof like the one I gave above for a more general type of equation, and some other results without proof (including Joe Silverman's) is this paper by Goldfeld math.columbia.edu/~goldfeld/ABC-Conjecture.pdf –  quid Sep 12 '12 at 13:51
Here is a survey article by Andrew Granville and Tom Tucker: ams.org/notices/200210/fea-granville.pdf –  Micah Milinovich Sep 14 '12 at 12:27

## protected by Todd Trimble♦Aug 2 '14 at 23:10

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