A Possible Breakthrough in Explaining a
Mathematical Riddle
By KENNETH CHANG
Numbers, addition, multiplication -- the basic stuff of
grade-school arithmetic -- are suddenly the excited talk of
cutting-edge mathematicians.
On Aug 30, with no fanfare, Shinichi Mochizuki, a
mathematician at Kyoto University in Japan, dropped onto the
Internet four papers.
The papers, encompassing 500 pages and four years
of effort, claim to solve an important problem in number theory
known as the abc conjecture. (No, it does not involve the alphabet;
it has to do with integers and prime numbers, and the letters
represent mathematical variables used in equations.)
He has remained quiet since then. Others have
not.
"I hope it's right," said Minhyong Kim, a
mathematician at the University of Oxford in England. ''It would be
a fantastic breakthrough.''
What is even more fascinating is that Dr. Mochizuki
has devised new mathematics machinery that he employs for the
proof.
The abstract ideas and notations that
mathematicians manipulate are unfathomable to most people. Dr.
Mochizuki's new mathematical language -- on his Web page, he
describes himself as an ''inter-universal geometer'' -- is at
present incomprehensible even to other top mathematicians.
"He's taking what we know about numbers and
addition and multiplication and really taking them apart," Dr. Kim
said. ''He creates a whole new language -- you could say a whole
new universe of mathematical objects -- in order to say something
about the usual universe.''
Jordan Ellenberg, a University of Wisconsin
mathematician who writes a mathematics blog, Quomodocumque, said,
"At first glance, it feels like you're reading something from outer
space."
Some wonder whether the accolades are coming too
soon.
For other recent mathematical tours de force -- the
proofs of Fermat's last theorem, by Andrew Wiles at Princeton in
1995, and the Poincaré conjecture, by Grigory Perelman, a Russian
mathematician, in 2003 -- other experts could not immediately tell
whether the proofs were valid, but "at least in some outline
version, they understood how this approach made sense," said Nets
Katz, a mathematician at Indiana University
For Dr. Mochizuki's abc conjecture proof, "that
seems to be completely missing, and I've never seen that in my
life," Dr. Katz said. "It just seems a little odd that most of the
people who say positive things about it cannot say what are the
ingredients of the proof."
While they cannot yet make heads or tails of it,
many are nonetheless taking it seriously, because Dr. Mochizuki
already has a number of significant proofs to his credit. "He has a
long track record, and he has a long track record of being
original," Dr. Ellenberg said.
Indeed, much of the buzz is around the new
techniques the mathematicians do not understand, potentially useful
in unraveling similar problems and revealing deeper connections
between numbers and geometry.
"It's a very basic restriction on how
multiplicative properties of numbers can interact with addition,"
Dr. Kim said about the abc conjecture. "Somebody might say: 'Could
there be anything new to discover about the relation between
addition and multiplication? This is grade-school stuff.' But there
is."
The conjecture, first proposed in the 1980s,
considers a simple equation of three integers, a + b = c, where the
three integers do not share any common divisors other than 1. For
example, 1 + 2 = 3 and 81 + 64 = 145 satisfy that condition (but 2
+ 2 = 4 does not). The conjecture also ties in the notion of prime
numbers -- integers greater than 1 that have no positive divisors
other than 1 and themselves.
"The conjecture says roughly that if there are
prime numbers that divide either a or b too many times, then their
presence has to be 'balanced out' by largish primes that divide c
only a few times," Dr. Kim said. "We see 3 divides 81 four times,
and 2 divides 64 six times. But then, 145 equals 5 times 29, so you
get the larger primes 5 and 29 dividing 145 just once."
As mathematicians learn Dr. Mochizuki's new
language, the proposed proof will take months or years of careful
review, not just the 500 pages that Dr. Mochizuki just released but
also reams of earlier work over the past dozen years.
"I'm very tentative about almost anything I say
about it, because it's such a long and technical paper," Dr. Kim
said. "It will take a long time to say something intelligent. It
will take a while before we even have a general picture of the
situation."
Science Times, The New York Times, September 18, 2012 Pg
D3