I was invited by the Computational Anatomy group at the Medical
Imaging Analysis Lab to see a PBS documentary on
Fermat’s Last theorem. Fermat, a lawyer by profession, was also a
prolific mathematician. He left behind one of the hardest problems in
mathematics.
Mathematically, his theorem stated that for the equation:
is unsolvable for the case n>2 (a, b, c and n are integers.) For the
n=2 case, the equation is simply the Pythagoras theorem. He also left
behind a note which said that he had a proof for statement, but the
margin was too small to contain it.
The solution to the problem was done by Dr. Andrew Wiles from
Princeton using 20th century mathematics. It’s highly unlikely that
Fermat had thought of the same proof.
The key to his proof were in the connections between elliptical curves
and modular forms. Elliptical curves are equations of the form:
The equations are non-singular (don’t intersect.) Cryptography based
on elliptical curves have been proposed to replace RSA because the
problem is fully exponential, unlike prime factoring which is
sub-exponential.
Back to our story…
Right after the second world war, two Japanese mathematicians Shimura
and Taniyama proposed a conjecture that elliptical curves are just
modular forms in disguise (known as the Shimura-Taniyama
theorem.) Think of modular forms as highly symmetric functions.
Decades passed without anyone making any breakthroughs in proving the
Shimura-Taniyama conjecture, but that didn’t stop people from
developing math atop it. Then, in the mid-80s, Dr. Kenneth Ribet
from UC Berkeley made a connection between the conjecture and Fermat’s
theorem.
His argument is as follows: If somebody were to find a set of numbers
that satisfies Fermat’s equation, then this set of numbers could be
used to construct an elliptical curve that is not modular, thus
disproving the Shimura-Taniyama conjecture.
Working backwards from the graphic, if the Shimura-Taniyama conjecture
is true, then all elliptical curves are modular, thus with no solution
to Fermat’s equation, making it true.
The rest of the documentary was about how Dr. Wiles took on the
challenge to prove the Shimura-Taniyama conjecture. Solving Fermat’s
last theorem had been his life-long ambition. He worked for many years
in isolation and when he finally proposed a solution, it was found to
have a mistake. Disappointed, he re-traced his steps and proposed an
alternative argument to the solution. He had solved the hardest
problem in math and become a part of history.
I think this documentary was more about ambitions, dreams and
aspirations of a person than the mathematics. Dr. Wiles took upon this
monumental task of achieving his childhood dream. Fame, money and
prestige were unimportant to him.
In our discussion that ensued, it was clear that major breakthroughs
can no longer happen by one person alone. One of the scientists in the
documentary read out a list of at least 30 people who directly
contributed to the proof.
