From Graham Hancock's blog

A Japanese mathematician claims to have the proof for the
ABC conjecture, a statement about the relationship between prime numbers that
has been called the most important unsolved problem in number theory.

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 twenty-first 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."

The conjecture has also been described as a sort of grand
unified theory of whole numbers, in that the proofs of many other important
theorems follow immediately from it. For example, Fermat's famous Last Theorem
(which states that an+bn=cn has no integer solutions if n>2) follows as a
direct consequence of the ABC conjecture.

In a 1996 article in The Sciences, the mathematician Dorian
Goldfeld of Columbia University said the ABC conjecture "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.

"No wonder mathematicians are striving so hard to prove
it – like rock climbers at the base of a sheer cliff, exploring line after line
of minute cracks in the rock face in the hope that one of them will offer just
enough purchase for the climbers to pick their way to the top."

And now, one such climber may have reached the summit.
According to Nature News, Mochizuki, a mathematician at Kyoto University, has
proved extremely deep theorems in the past, lending credence to his claim that
he has the proof for ABC. However, a huge investment of time by many other
mathematicians will be required to go through the gargantuan proof and verify
the claim.

"If the ABC conjecture yields, mathematicians will find
themselves staring into a cornucopia of solutions to long-standing
problems," Goldfeld wrote.

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