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Fermat's Last Theorem and the Fourth Dimension
Fermat's Last Theorem has got to be one of the most popular problems in the history of mathematics |millions of people have toyed with it, and thou- sands have worked up a real mental sweat trying to solve it. The problem, posed by the French mathematician Pierre de Fermat back in the seventeenth century, is usually stated in terms of the famous equation x n + y n = z n ; where x; y; z and n represent unspeci ed whole numbers. When n = 1 the equation has too many solutions to be interesting, and when n = 2 there are still in nitely many (3 2 + 4 2 = 5 2 is the most famous). The problem Fermat bequeathed to us is to show that when n becomes bigger than 2, the situation changes dramatically: there are no solutions at all. That is: when n is a whole number bigger than 2, no number that is the nth power of a whole number can be written as the sum of two smaller nth powers. It stands to reason that a proposition so tantalizingly simple would have a simple proof or a simple disproof. Yet for over three centuries the problem resisted the e orts of the sharpest minds that tackled it | and we still don't have a simple proof. Fermat's Last Theorem came to light after Fermat's death, when his son Clement-Samuel was cleaning up the old man's library. An especially cher- ished work in the elder Fermat's collection had been a seventeenth-century Latin edition of a millennium-old Greek treatise on numbers by the mathe- matician Diophantus of Alexandria. On one page, Diophantus discussed the problem of writing a given square as a sum of two squares; writing in the margin of that page, Fermat made his no-go claim about higher powers and famously said he'd found a wonderful proof of this result but couldn't include it because the margin was too small. The claim is not found elsewhere in Fermat's known writings, but on several occasions he did state that a third power can't be the sum of two smaller third powers, or a fourth power the sum of two fourth powers. How- ever, in the combative fashion of the times, Fermat would often announce his results indirectly, by proposing challenges for other mathematicians to test
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their
wits on. He thought that these challenges would give others a greater
appreciation of the hidden depths surrounding his problems about numbers
and lure them into doing active research on the topic, but sometimes the
tactic back red on him. For instance, in one of his letters he challenged
the English mathematician John Wallis to solve two problems:
1. given a cube,to write that cube as a sum of two cubes; and
2. given a sum of two cubes, to write that number as a sum of two cubes
in a di erent way.
The rst problem has no solution; in fact, this is just the case n = 3
of Fermat's Last Theorem. The second problem has many solutions; for instance,
(3=2) 3 + (5=3) 3 can also be written as (2) 3 + (1=6) 3 (Fermat was concerned
here with fractions as well as whole numbers). What Fermat seems to have
wanted was for Wallis to demonstrate that the rst problem had no solutions
and then to give a systematic approach to solving the second problem.
That is the real two-part challenge Fermat had in mind. But from the way
Fermat wrote the challenge, the rst part seems to be asking Wallis to
do something that is in fact impossible, and that Wallis probably suspected
was impossible (perhaps after hours of fruitless work). It's understandable
that Wallis resented this sneaky way of disguising the nature of the challenge,
and later passed up few opportunities to disparage Fermat's work on numbers.
Although Fermat's e orts to interest his contemporaries in problems about
numbers were unsuccessful, he did nd followers posthumously, start- ing
in the century after his death. It was up to these disciples to ll in
the blanks in Fermat's work, since Fermat himself had been loath to write
down details. By the middle of the nineteenth century, all of Fermat's
many claims had been proved (or, in a case or two, disproved), with the
exception of the famous marginal note. This \Great Theorem of Fermat"
also acquired the name \Fermat's Last Theorem" to mark its recalcitrance.
Nowadays many people call it \FLT" for short. Ironically, everyone knew
how the proof of FLT should begin: \Suppose there did exist non-zero whole
numbers x; y; z satisfying x n + y n = z n , with n >2. Then ...." In
mathematics, to prove that something doesn't exist, it's
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frequently helpful to assume for argument's sake that the thing does exist, and then show that the thing, merely by existing, would have to possess mutually incompatible properties, thus demonstrating that it couldn't exist in the rst place. This is the method of proof by contradiction, or reductio ad absurdum, and it's the method of choice for a problem like this. So, people knew what the seed of the proof should be, but there has to be some sort of soil into which a seed can be planted. Fermat himself, back in the seventeenth century, seems to have tried planting the seed in the obvious place: the study of the properties of ordinary whole numbers. This study nowadays is called elementary number theory (to distinguish it from the more abstract developments that came later). Leonhard Euler, who as the rst of Fermat's posthumous disciples revived the study of numbers in the eighteenth century, was able to construct proofs of FLT for the cases n = 3 and n = 4 (proofs conceivably found earlier by Fermat), using elementary methods. But going beyond n = 3and n = 4was hard. Euler's successors, and their successors up till the middle of the nineteenth century, were able to settle a few more cases, but this approach petered out and couldn't even be made to handle a value of n as small as 11. It seems that the ground of elementary number theory just doesn't have the right sort of nutrients for the seed of the proof of FLT | the kernel of contradiction | to sprout and grow into a full and rigorous argument. In the middle of the nineteenth century, mathematicians like Ernst Ed- uard Kummer found a di erent plot of land to plant the seed in: a new sub-discipline within number theory called algebraic number theory, and a sub-sub-discipline called the theory of cyclotomic number rings. Cyclotomic number rings are extensions of the ordinary arithmetic of whole numbers, in which other sorts of numbers, including imaginary numbers like the square root of minus one, are brought into the game. With the new methods, it became possible to prove FLT for many more exponents. Kummer more or less settled FLT for all exponents under 100 (he made a few mistakes on the hard ones). When Kummer's mistakes were corrected and his methods were extended and married with the power of twentieth-century computers, it became possible to prove FLT for all ex- ponents up into the low millions. But, for all mathematicians knew, these corroborations were a uke; FLT might have been false not just for one exponent, but for in nitely many exponents | perhaps even for all prime exponents with more than a million digits.
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Someday mathematicians might know enough about cyclotomic number rings to be able to construct a proof along the lines that Kummer envisioned; but it seems that the soil of algebraic number theory, in its current state, doesn't have the right nutrients either. Over the course of much ofthetwentieth century, professional interest in Fermat's Last Theorem as a hot research topic dwindled. The problem was still part of the lore of mathematics, and part of the eld's long-term agenda, but mathematicians found it hard to come up with new plans of attack that hadn't already been tried. No one had an idea how to proceed with FLT, and some experts even began to suspect that Fermat might have guessed wrong. But outside the academies, more people were working on the problem than ever before. Amateurs were attracted to the problem for a number of reasons. First, FLT is a simple and catchy question. Second, the fact that Fermat claimed to have found a proof raised people's hopes that a proof, indeed a simple proof, could be found. Third, there are certain people who are attracted to a problem precisely because it's hard, and here was a problem that a whole community of experts, the world's mathematicians, had despaired of solving with existing tools. Fourth, there was a cash prize for the person who solved the problem. And fth, it's easy to almost prove Fermat's Last Theorem, in a certain sense. Remember the basic strategy for proving FLT: you assume that it's false and derive a contradiction. Well, it's very easy to arrive at contradictions in mathematics | just make one mistake and, unless you inadvertently make another mistake that cancels out the rst one, you're likely to hit on two assertions that don't square with each other. Even if you nd your mistake, or it's pointed out to you, and you realize that your attempted proof by contradiction isn't valid, it's easy to convince yourself that, since you found a proof of FLT with only one mistake in it, you might be close to nding a proof with none. This psychological e ect made Fermat's challenge a very addictive problem to work on. But despite the serious e orts of very many people, with various degrees of persistence, no one could nd a proof. Finally, in the last decade of the twentieth century, mathematician An- drew Wiles, aided by his former student Richard Taylor, gave aproofofFer- mat's Last Theorem. The proper soil for the seed, or at least one proper soil for it, had been found: an area called the theory of elliptic curves, whose bor- ders Fermat himself had rambled across but whose true shape didn't emerge until the nineteenth century, and whose central inner jungle, still untamed
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