Kyoto University mathematician Shinichi Mochizuki has published a lengthy proof that may put to rest what many consider the most important unresolved problem in modern mathematics.
Mochizuki’s 500-page proof, first published on the internet in August, claims to provide proof for the abc conjecture in number theory.
The abc conjecture is also known as the Oesterlé-Masser conjecture because it was first proposed by David Masser in 1985 and Joseph Oesterlé in 1988 as an integer analogue of the Mason-Stothers theorem for polynomials.
The conjecture is stated in terms of three positive integers, a, b and c which have no common factor and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is rarely much smaller than c.
The abc conjecture is deemed important because it provides immediate proofs for a number of interesting theorems including Fermat’s last theorem, a vexing issue in number theory that took roughly 350 years to be demonstrated.
If Mochizuki’s proof withstands scrutiny, it will represent “one of the most astounding achievements of mathematics of the 21st Century,” said Dorian Goldfeld, a mathematician at Columbia University in the September 10 online edition of Nature magazine. In 1996 Goldfeld called the abc conjecture as “the most important unsolved problem in Diophantine analysis”.
Mochizuki, 43, calls the theory on which his proof is based the “inter-universal Teichmüller theory”. It can be used to prove Szpiro’s conjecture and Vojta’s conjecture, among other problems.
To support his theory Mochizuki has devised “techniques that very few other mathematicians fully understand”, said Nature. The proof “uses a huge number of insights that are going to take a long time to be digested by the community,” said Stanford professor Brian Conrad.
“[Mochizuki] has a long track record, and he has a long track record of being original,” Jordan Ellenberg, a University of Wisconsin mathematician, told the New York Times on Monday.
“Professor Mochizuki is an able mathematician and there is a great possibility of him having proved the abc conjecture this time,” said University of Tokyo professor Yujiro Kawamata.
“He spent at least 10 years in writing the thesis with uninterrupted focus throughout on it,” said a researcher acquainted with Mochizuki.
Shinichi Mochizuki was born in Tokyo in 1969. He was 5 when his family moved to the US to follow his father’s career. At the age of 16 Shinichi began studying at Princeton. At the age of 19 he graduated with a BS in mathematics. He was 23 when he began teaching as an assistant professor at Kyoto University in 1992. At that time he wasn’t yet fluent in Japanese, having spent most of his life in the US.
Mochizuki became a full professor at the age of 32. Since then he has become recognized as an important thinker in number theories. In 2005 he was honored by Japan Academy as one of the most important young scholars aged 45 and under.
