Fermat’s Last Theorem
(This article draws on the excellent paperback, Fermat’s Enigma” by Simon Singh.)
I begin with a riddle:
Three Navajo women sit side by side on the ground. The first woman, who is sitting on a buffalo skin, has a son who weighs 70 pounds. The second woman, who is sitting on a deer skin, has a son who weighs 80 pounds. The third woman who weighs 150 pounds is sitting on a hippopotamus skin.
The moral: the squaw of the hippopotamus is equal to the sons of the squaws of the other two hides.
If you were paying attention in the 8th grade, you know that the square of the hypotenuse equals the sum of the squares of the other two sides. Z^2=x^2 +y^2
Fermat’s Last Theorem states that for z^n=x^n +y^n, there are no whole number solutions for n>2. He wrote in the margin of his copy of Diophantus’ “Arithmetica” , originally published around 300 AD, 6 of the original volumes survived the Dark Ages; Fermat was using a French translation published in 1621.
He made a marginal note, circa 1637, when he was 36: “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” Fermat was known among mathematicians as a tease. At least one of his puzzlers was proven wrong. For those striving for cultural awareness, not mere dilettantes, his full name is Pierre de Fermat, the de denoting royalty not widowhood.
There are an infinite number of solutions for the case n=2, e.g. 3, 4, 5 and 5, 12, 13. You can generate your own by selecting any two whole numbers p and q, where p> q. p^2+q^2, p^2 q^2, and 2pq are Pythagorean triples.
Let’s do some cubing. Try 7, 6, and 5. 7 cubed is 49×7=343. 6 cubed is 216 and 5 cubed is 125, which total 341, not 343. Come on, you say, isn’t that close enough, an error of less than 1%. NO.
Infinity is a mathematical term meaning for a set which is equivalent to a proper subset of itself. The even numbers are a proper subset of the natural numbers. Infinity is not all the grains of sand at all the beaches in the world, not all the hydrogen atoms in all the world’s water molecules, not how many times I’ve told you to clean up your room. Computers have confirmed Fermat’s theorem for all n up to 4 million. Nothing in this world is infinite. The word you should use is googol, a 1 followed by 100 zeros .The founders of Google are good at sorting, not at spelling. Their corporate headquarters is known however, as the Googolplex, which is 10 raised to the googol power about a trillion trillion. You could count to a trillion in 31, 710 years.
All of the mighty mathematicians over the last 350 years have failed to completely prove the theorem. Leonard Euler proved it for n=3 utilizing the square root of -1. Sophia Germain, born April Fool’s day 1776, proved it for all prime numbers of the form 2p+1 where p is prime. She wrote to Gauss, the inventor of the bell curve as S. Germain. He said he wasn’t interested in Fermat’s theorem but encouraged her. In 1806 when Napoleon invaded Prussia, Germain asked her friend, a general in charge of the invading armies, to guarantee Gauss safety and he did. Gauss didn’t know who she was. Note that it wasn’t always Germany invading France.
In 1963, when Andrew Niles was 10, he read this book from his local library in England. He thought he would be first. I don’t think I went to the library until I was 12 and that was to read the “Freddy the Pig” books. At Oxford he did graduate work in elliptic curves. They’re very complicated, which is my way of saying “I don’t know what they are”.
In 1954 in Japan was born the Taniyama- Shimura conjecture that there was an equivalence between elliptic curves and modular forms. This turned out to be key to Niles solution. Taniyama had several peculiar aspects: He couldn’t tie his shoe laces and he always wore the same suit made of a green fabric with a strange metallic sheen. No one else in his family wanted that bolt of cloth. In 1958 he fell in love and got engaged, and soon after he committed suicide. His suicide note said he didn’t know why he was killing himself. A few weeks later his fiance killed herself.
In 1983 Gerhard Frey of Germany conjectured that proving the Taniyama- Shimura conjecture would prove Fermat. In 1987 the American Ken Ribet proved Frey. In 1986 Niles started. He worked in isolation in his attic. He married and had two children while he was pondering, so he must have come down from the attic occasionally, or she went up to the attic to save time.
After 6 years, he decided he had to talk to someone. He chose Ken Katz , a colleague at Princeton. He was so afraid of others knowing what he was doing that he created a course for graduate students called Calculations on Elliptic Curves, which Katz audited. It wasn’t long before it was just Katz. After 7 years, he was done.
Niles decided to go public with his proof, which was 200 pages long, at a seminar at the Isaac Newton Institute at Cambridge. One of the organizers was Niles thesis advisor, Coates. So he got three one hour time slots to do his proof. This was May 1993 and the world shook with the sensational news.
There was a disturbing sequel. To be an official proof it had to be published and refereed. Not just 2 or three referees, but six, one of whom was Nick Katz, who got chapter 3 which was 70 pages long. He got a helper, too. He spent the summer poring over it, checking with Niles about any doubts. Katz found an error and it took Niles another year to correct it, using a different method. This was covered in the pseudo course and he didn’t catch it then. He didn’t want to interrupt too much. After another year, Niles recast the proof (with the help of one of his former students).
Isaac Newton said, “If I have seen further it is by standing on the shoulders of giants.” Niles should have said the same. What he did say, after his three hour presentation was ” I think I’ll stop here”.
