4.8.05

A HIGHER STANDARD OF AMATEUR. I'm going to hold off on working through and commenting on Paulo Ribenboim's Fermat's Last Theorem For Amateurs. It is not the kind of book that one can read in bed or on the lounger with some music or talk radio playing; it helps to have a pencil handy to make marginal notes, as well as a pad of paper to tackle some of the more challenging bits. (Perhaps if I make my 50, this will be No. 51.)

Let me give you a feel for what's involved. From the preface:

The following general remarks are quite obvious and henceforth will be taken for granted.

If n is odd then Xn + Yn = Zn has a non-trivial solution if and only if
Xn + Yn + Zn = 0 has a non-trivial solution.

If x,y,z are non-zero integers such that xn + yn = zn, if d = gcd(x,y,z) and x1 = x/d, y1 = y/d, z1 = z/d then x1n + y1n = z1n, where the non-zero integers x1, y1, z1 are pairwise relatively prime. So, if we assume that Fermat's equation has a non-trivial solution, then it has one with pairwise relatively prime integers.

It goes on in a similar vein. First, there's some instructive material on Pythagorean triples. There's an algorithm for producing Pythagorean triples. Choose any two integers a, b, satisfying a > b > 0; then define x = 2ab, y = a2 - b2, and z = a2 + b2. But we're not interested in simply producing all the Pythagorean triples (a countable infinity; use the same method you'd use to count the rationals.) It's the primitive solutions we're after; namely, those for which gcd(a,b,c) = gcd(x,y,z) = 1. That makes possible the development of Fermat's first result, X4 - Y4 = Z2 has no solution in integers all different from zero. I want to work through all this as Ribenboim addresses something that once occurred to me: perhaps the last theorem is Fermat's method of infinite descent, which does work for even powers, although (and perhaps he discovered this later) it fails for powers that are odd primes. Whew. Bedtime, and some lighter reading.

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