How is Pythagoras (500 BC) linked to Galois (1800’s) or Wiles(1900’s)? I recently watched a terrific documentary about Sir Andrew
Wiles. The video follows Wiles’ quest to
prove Fermat’s Last Theorem, which states that there is no whole number solution to x^n+y^n=z^n greater than 2 (a cousin to the famous Pythagorean
Theorem). Fermat claims that he did not have room to
write the full proof in the margin of his copy of the

*Arithmetica.*Over the centuries no one, not even brilliant mathematicians like Gauss, Kummer, Euler, or Germain, has been able to prove what Fermat claimed as true. Up until 1993, even with the advantage of computer assisted algorithms, thousands of people had tried to prove it but no one had succeeded.
Wiles’ dream began when he was only ten years old, after
reading a math book in his local library and understanding that Fermat’s 300
year old theorem could be proven. He worked tirelessly for many
years - researching and contemplating
the works of many mathematicians, bridging seemingly unconnected areas of math,
and handwriting all of his thoughts on paper and chalkboard. Using ideas, theories, and proofs from mathematicians
including Taniyama, Galois, Iwasawa, and Flach, in order to formulate his own
conjectures and theories, Wiles finally revealed his proof of Fermat’s Last
Theorem in 1993. However, several months
later, there was a problem discovered with the last part of the proof. Wiles went back and started picking his proof
apart to see where he went wrong, only to find that the dots could be connected
using a path he had previously abandoned. So in the end, Wiles really did
uncover a proof to Fermat’s Last Theorem.
One of Wiles’ great accomplishments on this journey was the webbing
together of so many areas, ideas, and branches of math. I made the word cloud (below) which contains
some of the theories and mathematicians whose ideas Wiles used to uncover his
proof to Fermat’s Last Theorem.

Quote from Documentary –

ANDREW WILES: “Perhaps I could best describe my
experience of doing mathematics in terms of entering a dark mansion. One goes
into the first room, and it's dark, completely dark. One stumbles around
bumping into the furniture, and gradually, you learn where each piece of
furniture is, and finally, after six months or so, you find the light switch.
You turn it on, and suddenly, it's all illuminated. You can see exactly where
you were. At the beginning of September, I was sitting here at this desk, when
suddenly, totally unexpectedly, I had this incredible revelation. It was the
most—the most important moment of my working life.”

Never heard of Kummer!

ReplyDeleteI know the content is intense, but you might either add some idea of how he proved it (most general terms) or what his achievement means to math as a profession. Or what FLT being an unsolved problem for so long means about math or meant to mathematicians.