Higher mathematics ...

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Fermat

In the movie Bedazzled, our hero Elliot Richards (played by Brandon Fraser) is tempted by the devil (Elizabeth Hurley) to choose any seven wishes. Of course, our bumbling hero signs on and the romp through ridiculous humor is on. In one part of the film the devil plays a teacher for a group of school boys and "blows them away" by not making any assignments. But it is the math assignment that is most interesting. The math homework question is to solve the equation
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for integers $x,\ y,\ $and $z.$ This equation appears to be quite similar to the Pythagorean theorem with higher exponents. Had the movie been made just 10 years earlier, this would have constituted a 360 year old unsolved problem known as Fermat's Last Theorem. It is named after Pierre de Fermat (1601-1665) who discovered it.

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Galois
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Gauss
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Kummer

Over the last three centuries this theorem has tempted some of the very greatest mathematicians that have ever lived. None were successful. One of the first women mathematician superstars, Sofie Germain (1776-1831), made some headway on the problem but could not carry out the whole proof.
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In 1994, the Princeton mathematician Andrew Wiles, after seven years of concentrated effort, produced a proof that shocked the world. His proof used little of the historical scafolding, and relies on the modern theory of elliptic curves and other subjects. Said Wiles, "I was first attracted to this problem when I was just 10 years old. It seemed so easy to understand."

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Wiles - about 10 yrs old
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Wiles

So you want to understand the proof? Here's how. First study Rotman's Homological Algebra and Shafarevich's Algebraic Geometry. Master also Apostol's Intro to Analytic Number Theory, Vol 2. (for the elementary stuff about modular forms). Now study Koblitz's book on Elliptic Curves and Modular Forms and Silverman's two books on Arithmetic of Elliptic Curves as well. Add to this mix Serre's "A Course in Arithmetic" for an understanding of p-adic rings/fields. You will also need to know Mazur's Ring Deformation theory and Hecke Algebras - no books available, so read the literature.

Keep in mind that Wiles taught a course on some of the stuff in the proof and his grad students all dropped out; it was too tough.

No homework on this subject today. But, you might try the following theorems proved by Fermat.

  1. (Fermat's Little Theorem.) If $p$ is any prime and $a$ any is positive integer then $p$ divides $a^{p}-a$.

  2. If $s$ is composite, then $2^{s}-1$ is composite.

  3. The only possible divisors of $2^{p}-1$ have the form $2pk+1.$

  4. (Goldbach Conjecture.) Every even integer can be written as the sum of exactly two primes.

This last problem is extremely difficult; it remains open and beyond the reach of all attempts.

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