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Proof claimed for deep connection between primes : Nature News & Comment. The usually quiet world of mathematics is abuzz with a claim that one of the most important problems in number theory has been solved. Mathematician Shinichi Mochizuki of Kyoto University in Japan has released a 5.
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Diophantine' problem. The abc conjecture, proposed independently by David Masser and Joseph Oesterle in 1. Fermat’s Last Theorem, but in some ways it is more significant. It involves the concept of a square- free number: one that cannot be divided by the square of any number.
Fifteen and 1. 7 are square free- numbers, but 1. 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 n that are prime numbers. For instance, sqp(1. It concerns a property of the product of the three integers axbxc, or abc — or more specifically, of the square- free part of this product, which involves their distinct prime factors. It states that for integers a+b=c, the ratio of sqp(abc)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=1.
In this case, in which r=2, sqp(abc)r/c is nearly always greater than 1, and always greater than zero. It turns out that this conjecture encapsulates many other Diophantine problems, including Fermat’s Last Theorem (which states that an+bn=cn has no integer solutions if n> 2). Like many Diophantine problems, it is all about the relationships between prime numbers.
According to Brian Conrad of Stanford University in California, “it encodes a deep connection between the prime factors of a, b and a+b”. Many mathematicians have expended a great deal of effort trying to prove the conjecture. In 2. 00. 7, French mathematician Lucien Szpiro, whose work in 1. Like Szpiro, and also like British mathematician Andrew Wiles, who proved Fermat’s Last Theorem in 1. Mochizuki has attacked the problem using the theory of elliptic curves — the smooth curves generated by algebraic relationships of the sort y. There, however, the relationship of Mochizuki’s work to previous efforts stops.
He has developed techniques that very few other mathematicians fully understand and that invoke new mathematical . The proof is spread across four long papers. And he adds that the pay- off would be more than a matter of simply verifying the claim.
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