In 2012 Japanese mathematician Shinichi Mochizuki claimed to have created a proof for a notoriously complex problem called the ABC conjecture, but no one could understand it.
Now, according to Japanese news site The Asahi Shimbun, the proof may soon be accepted for publication in a mathematical journal of the Kyoto University's Research Institute for Mathematical Sciences.
First proposed in the 1980s, the ABC conjecture concerns the relationship between positive integers that satisfy the statement a + b = c, but which also allows a fourth variable, d, to represent the distinct prime factors of abc in certain specified circumstances.
The puzzle, which ties together elements of both number theory and arithmetic geometry, has never been conclusively solved by mathematicians.
In August 2012, the celebrated Japanese mathematician unveiled a 500-page proof spread over four papers, which together purported to offer a revolutionary solution to a gruelling math problem mentioned earlier in the text. Mochizuki proposed a whole new type of mathematics called inter-universal Teichmüller theory (IUT).
A mathematician will soon publish his answer to an infamous conjecture, but it's so complicated no one can check if he's right.
In an effort to untangle the mathematics, mathematicians have held workshops and produced a summary paper, totalling a meagre 400 pages. But while some have been converted to IUT, others have remained sceptical of the proof, New Scientist reported.
“A small number of those close to Mochizuki claim to understand the proof, but they have had little success in explaining their understanding to others,” wrote Peter Woit at Columbia University in a blog post.
Mochizuki is universally described as a reclusive prodigy, and according to a Japanese newspaper The Asahi Shimbun, he spent a decade working in isolation to develop his theory and the proof could now be published as early as January next year.
“There has always been a rumour that the papers were submitted to a Japanese journal, which people were concerned would not give the papers enough scrutiny,” says Felipe Voloch at the University of Canterbury, New Zealand.
Though journal of the Kyoto University's Research Institute for Mathematical Sciences is a good reputable one, the fact that it is in Japanese, from Mochizuki’s own institution, and he is the journal’s editor-in-chief means questions will still remain.
“For me, the fact that it has been accepted in this journal doesn’t change much. I am still waiting for an explanation of the ideas that I can understand,” says Voloch.
Ivan Fesenko at the University of Nottingham, UK, disagrees, saying that “triple efforts” would have been applied to make sure that everything was fine before publication. And that the choice of journal can be explained by the fact that the top mathematicians in the field are Japanese.
“This is an achievement on the scale we see very-very rarely in mathematics. Essentially, this is the best result in number theory in the last 50 years,” Fesenko says.
So it seems the proof remains in a precarious position. The few mathematicians who say they understand the proof will continue to champion it, whilst others will remain sceptical, New Scientist states.
“Until there are either mathematicians who both understand the proof and are able to explain it to others, or a more accessible written version of the proof, I don’t think this proof will be accepted by the larger math community,” wrote Woit.
There’s a general sense among non mathematicians that the subject is either right or wrong, and the truth is easily discovered. While mathematics does insist on rigorous, logical proof of correctness, mathematicians often argue over the details. This is good for study since it generally leads to better exposition and streamlined proofs.
These arguments have happened before. Wiles’ proof of Fermat’s Last Theorem was scrutinized thoroughly, and an error was found which had to be corrected. Perelman’s work on the Poincaré Conjecture was only a detailed sketch of a proof which required hard work on the part of others to be made rigorous. Mochizuki’s work may eventually pass the test, but it could take many years before we get to a clean version that can be more widely understood.
What's your opinion on the subject - are you sceptic or converted to IUT?