There’s been a lot of buzz lately about Shinichi Mochizuki’s proposed proof of the ABC conjecture, a conjecture in number theory named after the variables used to state it. Rather than explaining the conjecture here, I recommend a blog post by Brian Hayes.

The ABC conjecture has been compared to Fermat’s Last Theorem (FLT). Both are famous number theory problems, fairly easy to state but notoriously hard to prove. (FLT is easy to state. ABC takes a little more work, but is accessible to a patient teenager.) And both have been proved recently, assuming the ABC proof holds up. But here are three contrasts between ABC and FLT.

- FLT was proposed in 1637, proved in 1995. ABC was proposed in 1985, possibly proved in 2012.
- The
*conclusion* of FLT is not that important, but the *proof* is very important. The conclusion of ABC is important, and nobody knows about the proof yet.
- The FLT proof established deep connections between widely-known areas of math. The proof of ABC is comparatively self-contained, relying on a new specialized area of math few understand.

I’m not an expert in number theory, not by a long shot, but I don’t believe many proofs cite Fermat’s Last Theorem *per se*. Instead, there are proofs that depend on the more abstract results that Wiles proved, results that imply FLT as a corollary. And long before Wiles, a tremendous amount of math was motivated by attempts to prove FLT.

The ABC conjecture is a more technical statement, as its name might imply. The conjecture itself has wide-ranging applications, if it is true. The proof may also be important, but apparently nobody knows yet. The proof of FLT brought together a lot of existing machinery, but the ABC proof created a lot of new machinery that few understand.

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Its worth pointing out that ABC implies FLT for all sufficiently large exponents n.

My number theory professor had this to say about FLT (before it was proven):

“It is a great theorem. It has destroyed so many great mathematicians trying to prove it…”

I’m also not aware of any mathematical papers which cite FLT. But

lotscite the Taniyama-Shimura conjecture (now called the Modularity Theorem).That supports the statements Cook made. In fact, earlier a mathematician named Frey had connected FLT to elliptic curves. I believe that Serre and Ribet then showed that FLT would be a consequence of Taniyama-Weil (that is what I call it). Wiles proved Taniyama-Weil. So T-W is the result that has consequences.