October 11, 2012
With the presidential election, tension with Iran, and trouble in Libya and Syria dominating the news this fall, it’s understandable that Kyoto mathematician Shinichi Mochizuki’s claim to have proved the ABC conjecture (reported in the New York Times) may not be getting the attention that it deserves.
While topics of broader interest occupy dinner table conversations, among practitioners of the subfield of pure mathematics known as number theory, Mochizuki’s work is the hot topic. If his proof holds together, it’s possible that 2012 will be remembered into the distant future primarily as the year that ABC was proved.
The only problem is that Mochizuki’s work is so esoteric that it’s proving difficult for the mathematical community to check his proof.
The ABCconjecture, proposed by Joseph Oesterle and David Masser in the 1980’s, is a technical assertion about the prime divisors of three numbers, called a,b,and c, that satisfy a+b=c. It’s interesting because it implies the truth of a whole host of other difficult problems in number theory, solving all of them at once.
It might seem remarkable that we don’t already know everything there is to know about such a simple equation, but in fact there are many unanswered questions about the relationships between multiplicative and additive properties of whole numbers. Two famous unsolved problems are the twin prime conjecture, which asserts that there are infinitely many pairs of consecutive odd prime numbers (like 11 and 13, 17 and 19, 29 and 31, and so forth), and Goldbach’s conjecture, which asserts that every even number is the sum of two primes.
Unlike ABC, these two problems are more curiosities than problems of central theoretical importance.
Read more posts by Jeremy Teitelbaum, dean of the College of Liberal Arts and Sciences, on his blog.