Stephen Karlson (shkarlson) wrote in 50bookchallenge,
Stephen Karlson

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Book Review No. 25 is Amir Aczel's Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem.  The book is short, readable, and full of all sorts of mathematics lore, although "unlocking the secret" is still left to the reader as an exercise.  (The exercise is below the jump.)

Within the interesting mathematics lore are some of the usual favorites, including Gauss's short-cut for math busywork (you have fifty pairs of numbers summing to 100, with a 50 standing alone) and Euler's use of an equation to silence Denis Diderot.  There are also more obscure (to me, at least) results, including the Dirichlet prime number theorem, which establishes that if two integers a and b are relatively prime, the arithmetic progression a + b, a + 2b, a + 3b, ... includes a countable set of prime numbers.  The sum of any two such prime numbers, 2a + (m+n)b is an even number: might it be possible to account for all possible even numbers to crack the Goldbach conjecture?  That research is likely out there: perhaps it's a well-known dead end.

Where the reader has to do more work is in the guts of the proof of the Fermat conjecture.  The argument is straightforward enough:  elliptic curves are modular, therefore Diophantine equations of order greater than two have no nondegenerate solutions in integers.  Getting there is not half the fun.  First, those elliptic curves?  We're not talking about conic sections, rather (it's nomenclature borrowed from the classification of differential equations) an elliptic curve is a polynomial equation,

y2 = ax3 + bx2 + cx,

where the coefficients a, b, and c, can be integers or rational numbers, although there are transformations of variables to produce a simpler form,

y2 = x3 + ax + b.

There is additional mathematical magic establishing that the solution space of these curves is topologically equivalent to a torus.  Apparently the proof of the Fermat conjecture starts with a quest for a contradiction: if there is a Fermat triple an + bn = cn, where a, b, and c are integers and n is an odd prime, there is a corresponding elliptic curve

y2 = x (x - an) (x + bn),

that is not modular.  Perhaps there is no intuitive explanation for "modular."  Heck, that Wikipedia explanation is pretty heavy going!  Or perhaps, if there were a Fermat triple, you'd find it in the hole, rather than on the doughnut?

In any event, there was a lot of heavy lifting, and a lot of work in apparently unrelated areas of mathematics, to establish that there are no Fermat triples, and there may still be more than a few research papers waiting to be written.

(Cross-posted to Cold Spring Shops.)
Tags: academic, history, non-fiction, scholarly

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