Shinichi Mochizuki, a mathematician at Kyoto University in Japan, has released a 500-page proof of the “abc” conjecture, a celebrated unsolved problem originally posed in 1985.
Let sqp(n) denote the square-free part of an integer n, or in other words the product of the prime factors of n. For example, sqp(18) = 2 * 3 = 6 (here * denotes multiplication). The abc conjecture asserts that for integers a, b and c, where a + b = c, the ratio sqp(a*b*c)r/c always has some minimum value greater than zero for any value of r greater than 1. For example, if a = 3 and b = 125, so that c = 128, and r = 2, then sqp(a*b*c)=30 and sqp(abc)2/c = 900/128 = 7.03125.
It turns out that this conjecture encapsulates many other deep problems of number theory, including Fermat’s Last Theorem (which states that an + bn = cn has no integer solutions if n > 2), which was proven 17 years ago by Princeton mathematician Andrew Wiles. Brian Conrad of Stanford University notes that the abc conjecture “encodes a deep connection between the prime factors of a, b and a + b.”
Needless to say, Mochizuki’s long, multi-step proof must be carefully scrutinized by mathematicians in the field before it can be taken seriously. There have been many cases where a claimed proof of a major mathematical result was subsequently found to have serious flaws. Some general principles to follow in judging whether a mathematical result should be taken seriously are given in our previous Math Drudge blog.
For additional details on the abc conjecture and Mochizuki’s manuscript, see the informative Nature article, from which some of the above note was taken, Barry Cipra’s Science Now article, Kenneth Chang’s NY Times article and Jeremy Teitelbaum’s article in UConn Today. Additional background information may be seen in the Wikipedia article on the abc conjecture.