Originally posted by the Bear:
i wonder... how did Fermat do it?
all to prove that a^n = b^n + c^n cannot exist for n>2
it took Wiles stacks of paper, forays into Modular Mathematics and the Taniyama-Shimura conjecture...
yet Fermat said he could not show it in the margin of his book...
Wiles proved the Taniyama-Shimura conjecture using proof by induction.. which is:
(1) Prove that the statement is true for the first case
(2) Prove that if the statement is true for one case, it must be true for the next. (e.g. n+1)
A good analogy to this method would be the domino effect. First, you would have to get it rolling by knocking down the first domino. Then, you would have to make sure that the dominos are spaced such that any given domino would definitely hit the next one, causing a rippling “domino” effect, and thus proving all cases of a statement.
For this to happen, Wiles depended on GaloisÂ’s concept of group theory Galois in the 19th century had attempted to find a general solution to quintic equations using this method of group theory
i still don't understand group theory...
anyway..Wiles employed this method of group theory and was then able to prove that the first element of each E-series did indeed match the first element of each M-series. (E and M series derived from 5 clock arithmatic)
Wiles then ran into some problems trying to topple the rest of the dominoes. Eventually, he employed the Kolyvagin-Fach method, combind with the Iwaswa method to achieve this, and thus by proof of induction, solved the Taniyama-Shimura conjecture and thus FermatÂ’s Last Theorem.
i think fermat just anyhow say one la..then somehow got correct