Proven! — 'most important unsolved problem' in numbers

[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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[longship]

1. Wow! 500 pages of number theory!

Boggles the mind. I can't even wrap my brain around this and I taught mathematics for years. It is like a symphony of number theory.

Wonderful, esoteric shit here.
R&K

[drm604]

2. This is big if the proof is correct.

http://www.maa.org/mathland/mathtrek_12_8.html

Astonishingly, a proof of the ABC conjecture would provide a way of establishing Fermat's last theorem in less than a page of mathematical reasoning. Indeed, many famous conjectures and theorems in number theory would follow immediately from the ABC conjecture, sometimes in just a few lines.

"The ABC conjecture is amazingly simple compared to the deep questions in number theory," says Andrew J. Granville of the University of Georgia in Athens. "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."

"Nowadays, if you're working on a problem in number theory, you often think about whether the problem follows from the ABC conjecture," he adds.

"The ABC conjecture is the most important unsolved problem in Diophantine analysis," Goldfeld writes in Math Horizons. "It is more than utilitarian; to mathematicians it is also a thing of beauty. Seeing so many Diophantine problems unexpectedly encapsulated into a single equation drives home the feeling that all the subdisciplines of mathematics are aspects of a single underlying unity, and that at its heart lie pure language and simple expressibility."

[drm604]

3. The reference in the OP to data encryption raises a question in my head.

Does this have any impact on computational theory?

[eppur_se_muova]

4. No explicit mention of the Riemann Hypothesis ...

I realize "most important" is a matter of opinion, but still ...