Proven! — 'most important unsolved problem' in numbers [View all]

[Jim__]

From NBCNews

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If Shinichi Mochizuki's 500-page proof stands up to scrutiny, mathematicians say it will represent one of the most astounding achievements of mathematics of the 21st century. The proof will also have ramifications all over mathematics, and even in the real-world field of data encryption.

The ABC conjecture, proposed independently by the mathematicians David Masser and Joseph Oesterle in 1985 but not proven by them, involves the concept of square-free numbers, or numbers that cannot be divided by the square of any number. (A square number is the product of some integer with itself). According to the mathematics writer Ivars Peterson in an article for the Mathematical Association of America, the square-free part of a number n, denoted by sqp(n), is the largest square-free number that can be obtained by multiplying the distinct prime factors of n. Prime numbers are numbers that can only be evenly divided by 1 and themselves, such as 5 and 17.

The ABC conjecture makes a statement about pairs of numbers that have no prime factors in common, Peterson explained. If A and B are two such numbers and C is their sum, the ABC conjecture holds that the square-free part of the product A x B x C, denoted by sqp(ABC), divided by C is always greater than 0. Meanwhile, sqp(ABC) raised to any power greater than 1 and divided by C is always greater than 1. [ What Makes Pi So Special? ]

This conjecture may seem esoteric, but for mathematicians, it's deep and ubiquitous. "The ABC conjecture is amazingly simple compared to the deep questions in number theory," Andrew Granville of the University of Georgia in Athens was quoted as saying in the MAA article. "This strange conjecture turns out to be equivalent to all the main problems. It's at the center of everything that's been going on."

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