One of the world’s most complex mathematical theories may have finally been cracked by a scholar in Japan, although confirming the breakthrough may take many more years due to the sheer scale of the achievement.
Shinichi Mochizuki, a mathematician at Kyoto University, has released four papers on the internet describing his proof of what is known as abc conjecture.
The paper, which is 500 pages long, can be viewed on his website in a series of PDFs labelled “Teichmuller Theory”. The proof took four years to calculate and if confirmed it would be one of the greatest mathematical achievements of this century, experts said.
Confirming the breakthrough, however, may take just as long as Mochizuki has created an entirely new mathematical language to explain the steps that he took – and others in the field will have to learn to read it first. Since publishing his paper online on August 30, Mochizuki has declined to comment.
His peers are now trying to confirm the validity of the work, but it is agreed that 43-year-old Mochizuki has an excellent track record in advanced mathematics. The abc conjecture was first proposed by British mathematician David Masser, working with France’s Joseph Oesterle, in 1985. It was, however, never proven. It refers to equations of the form a+b=c. It involves the concept of a square-free number, meaning a number that cannot be divided by the square of any number. For example, 15 and 17 are square-free numbers, but 16 is not because it is divisible by 32. The “square free” part of a number, given the value n, is the largest square-free number that can be formed by multiplying the factors of n that are prime numbers. From that, the abc conjecture concerns a property of the product of the three integers abc, which involves their distinct prime factors. The conjecture states that for integers a+b=c, the square free part always has a minimum value greater than zero and nearly always greater than 1. Dorian Goldfeld, a mathematician at Columbia University in New York, told Nature magazine that Mochizuki’s discovery is “one of the most astounding achievements of mathematics in the 21st century.” On his web site, Mochizuki describes himself not as a mathematician but as a “inter-universal geometer” and – as long as his theory stands up to the scrutiny – there are hopes that his findings will settle a number of knotty problems in number theory and other branches of maths.
