Another milestone in the field of Maths!

Friday, September 21, 2012
A Japanese mathematician, Shinichi Mochizuki of Kyoto University, has claimed to crack one of the most complex mathematical theory of mathematics. He has released a 500-page proof of the abc conjecture - important problem of the number theory. It took him four years to propose this proof of the abc conjecture and The Telegraph has predicted that same amount of time might be required to verify his work!

abc conjecture was proposed by Joseph Oesterle and David Masser way back in 1980s. It is the most important unsolved problem in Diophantine analysis. It deals with the equations of the form c. It involves the concept of a square-free number: one that cannot be divided by the square of any number. Fifteen and 17 are square free-numbers, but 16 and 18 — being divisible by 42 and 32, respectively — are not. The 'square-free' part of a number n, sqp(n), is the largest square-free number that can be formed by multiplying the factors of that are prime numbers. For instance, sqp(18)= 2×3= 6.

Now, the abc conjecture is stated in terms of the product of the three integers c, or abc — or more specifically, of the square-free part of this product, which involves their distinct prime factors. It states that for integers c, the ratio of sqp(abc)r/always has some minimum value greater than zero for any value of r greater than 1. For example, if = 3 and = 125, so that = 128, then sqp(abc)= 30 and sqp(abc)2/c = 900/128. In this case, in which = 2, sqp(abc)r/c is nearly always greater than 1, and always greater than zero. 

Japanese mathematician Shinichi Mochizuki of Kyoto University

Many mathematicians before Mochizuki have tried to prove the conjecture. In 2007, French mathematician Lucien Szpiro, whose work in 1978 led to the abc conjecture in the first place claimed to have a proof of it, but it was soon found to be flawed. Mochizuki's proof is yet to be verified by other mathematicians which will require a long time. Mochizuki has called the theory on which his proof is based as Inter-universal Teichm├╝ller theoryDorian Goldfeld, a mathematician at Columbia University in New York told the Nature magazine, "The abc conjecture, if proved true, at one stroke solves many famous Diophantine problems, including Fermat's Last Theorem. If Mochizuki’s proof is correct, it will be one of the most astounding achievements of mathematics of the twenty-first century."

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