D. R. H.
Achilles and the Tortoise have come to the residence of their friend the Crab, to make the acquaintance of one of his friends, the Anteater. The introductions having been made, the four of them settle down to tea.
TORTOISE: We have brought along a little something for you, Mr. Crab.
CRAB: That’s most kind of you. But you shouldn’t have.
TORTOISE: Just a token of our esteem. Achilles, would you like to give it to Mr. C?
ACHILLES: Surely. Best wishes, Mr. Crab. I hope you enjoy it.
(Achilles hands the Crab an elegantly wrapped present, square and very thin. The Crab begins unwrapping it.)
ANTEATER: I wonder what it could be.
CRAB: We’ll soon find out. (Completes the unwrapping, and pulls out the gift.) Two records! How exciting! But there’s no label. Uh-oh-is this another of your “specials,” Mr. T?
TORTOISE: If you mean a phonograph-breaker, not this time. But it is in fact a custom-recorded item, the only one of its kind in the entire world. In fact, it’s never even been heard before—except, of course, when Bach played it.
CRAB: When Bach played it? What do you mean, exactly?
ACHILLES: Oh, you are going to be fabulously excited, Mr. Crab, when Mr. T tells you what these records in fact are.
TORTOISE: Oh, you go ahead and tell him, Achilles.
ACHILLES: May I? Oh, boy! I’d better consult my notes, then. (Pulls out a small filing card and clears his voice.) Ahem. Would you be interested in hearing about the remarkable new result in mathematics, to which your records owe their existence?
CRAB: My records derive from some piece of mathematics? How curious! Well, now that you’ve provoked my interest, I must hear about it.
ACHILLES: Very well, then. (Pauses for a moment to sip his tea, then resumes.) Have you heard of Fermat’s infamous “Last Theorem”?
ANTEATER: I’m not sure.... It sounds strangely familiar, and yet I can’t quite place it.
ACHILLES: It’s a very simple idea. Pierre de Fermat, a lawyer by vocation but mathematician by avocation, had been reading in his copy of the classic text Arithmetica by Diophantus and came across a page containing the equation
He immediately realized that this equation has infinitely many solutions a, b, c, and then wrote in the margin the following notorious comment:
has solutions in positive integers a, b, c, and n only when n = 2 (and then there are infinitely many triplets a, b, c, which satisfy the equation); but there are no solutions for n > 2. I have discovered a truly marvelous proof of this statement, which, unfortunately, is so small that it would be well-nigh invisible if written in the margin.
Ever since that day, some three hundred years ago, mathematicians have been vainly trying to do one of two things: either to prove Fermat’s claim and thereby vindicate Fermat’s reputation, which, although very high, has been somewhat tarnished by skeptics who think he never really found the proof he claimed to have found—or else to refute the claim, by finding a counterexample: a set of four integers a, b, c, and n, with n > 2, which satisfy the equation. Until very recently, every attempt in either direction had met with failure. To be sure, the Theorem has been proven for many specific values of n—in particular, all n up to 125,000.
ANTEATER: Shouldn’t it be called a “Conjecture” rather than a “Theorem,” if it’s never been given a proper proof?
ACHILLES: Strictly speaking, you’re right, but tradition has kept it this way.
CRAB: Has someone at last managed to resolve this celebrated question?
ACHILLES: Indeed! In fact, Mr. Tortoise has done so, and as usual, by a wizardly stroke. He has not only found a proof of Fermat’s Last Theorem (thus justifying its name as well as vindicating Fermat), but also a counterexample, thus showing that the skeptics had good intuition!
CRAB: Oh my gracious! That is a revolutionary discovery.
ANTEATER: But please don’t leave us in suspense. What magical integers are they, that satisfy Fermat’s equation? I’m especially curious about the value of n.
ACHILLES: Oh, horrors! I’m most embarrassed! Can you believe this? I left the values at home on a truly colossal piece of paper. Unfortunately it was too huge to bring along. I wish I had them here to show to you. If it’s of any help to you, I do remember one thing—the value of n is the only positive integer which does not occur anywhere in the continued fraction for π.
CRAB: Oh, what a shame that you don’t have them here. But there’s no reason to doubt what you have told us.
ANTEATER: Anyway, who needs to see n written out decimally? Achilles has just told us how to find it. Well, Mr. T, please accept my hearty felicitations, on the occasion of your epoch-making discovery!
TORTOISE: Thank you. But what I feel is more important than the result itself is the practical use to which my result immediately led.
CRAB: I am dying to hear about it, since I always thought number theory was the Queen of Mathematics—the purest branch of mathematics—the one branch of mathematics which has no applications!
TORTOISE: You’re not the only one with that belief, but in fact it is quite impossible to make a blanket statement about when or how some branch—or even some individual Theorem—of pure mathematics will have important repercussions outside of mathematics. It is quite unpredictable—and this case is a perfect example of that phenomenon.
Pierre de Fermat
ACHILLES: Mr. Tortoise’s double-barreled result has created a breakthrough in the field of acoustico-retrieval!
ANTEATER: What is acoustico-retrieval?
ACHILLES: The name tells it alclass="underline" it is the retrieval of acoustic information from extremely complex sources. A typical task of acoustico-retrieval is to reconstruct the sound which a rock made on plummeting into a lake, from the ripples which spread out over the lake’s surface.
CRAB: Why, that sounds next to impossible!
ACHILLES: Not so. It is actually quite similar to what one’s brain does, when it reconstructs the sound made in the vocal cords of another person from the vibrations transmitted by the eardrum to the fibers in the cochlea.
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