Arthur Clarke - The Last Theorem

And, like many other amateur mathematicians, before and after, Wolfskehl was immediately hooked. He forgot about killing himself, being too busy trying to plumb the mysteries of a-squared plus b-squared equals c-squared, and the paradox that if the quantities were cubed, they never did equal each other.

Then there was the also long deceased Sophie Germain, whose teenage years had been spent during the frightening time of the French Revolution. Why this should have persuaded young Sophie to resolve on a career in mathematics is not immediately obvious. But it did.

Of course, that was not an easy ambition for a female to accomplish. As Elizabeth I of England had once put it, Sophie was cursed by being split rather than fringed, and so everything she tried to do was vastly harder for her than for her fringed colleagues.

Then, when his imaginary conversation partners ran out of steam, something Myra de Soyza had said began to cudgel Ranjit’s mind.

What had it been? Something about seeing what tools other mathematicians had possessed at the time Fermat had jotted his cursed boast in the margin of his book?

Well, what tools were they?

He remembered that Sophie Germain was said to have been the first mathematician of any gender to make any headway at all with the Fermat proof. So just what headway had she made?

Ranjit, of course, had no way of looking that up. Back at the university, equipped with a password, all he would have had to do was hit a few keys on the handiest computer and the damn woman’s entire life production would have been laid out for him to study.

But he didn’t have the computer. All he had was his memory, and he was not sure that it was adequate to the task at hand.

He did remember what a “Sophie Germain prime” was—that is, any prime, p, such that 2p + 1 was also a prime. Three was the littlest Sophie Germain prime: 3 × 2 + 1 = 7, and seven was a prime, all right. (Most of the other Sophie Germain primes were much larger, and thus hardly any fun at all.) Ranjit was quite pleased with himself for remembering this, though no matter how much he thought about it, he could not see any way in which a Sophie Germain prime could lead him to a solution of the Fermat problem.

There was one other thing. After profound labor Germain had produced a theorem of her own:

If x, y, and z are integers, and if x 5+ y 5= z 5, then x, y, or z must be divisible by five.

Like every other stepping-stone toward a proof that Ranjit had managed to quarry out of the refractory stone of his mind, this one was a disappointment. The equation made no sense. Fermat’s whole theorem was supposed to prove that no such equality as x 5+ y 5= z 5could ever exist in the first place. So it wasn’t of any use at all….

Or was it? That is, forget Sophie Germain’s useless theorem itself, but how did she get to it?

And wasn’t that what Myra had suggested to him at Dr. Vorhulst’s party, back in the days when Ranjit could sometimes go to a party?

There was one other person (well, sort of person) with whom Ranjit had never, or never yet, had any personal dealings, but who (or which) could have supplied him with very useful data. It is probably about time that we spent a little more time with him (or them, or it, or maybe even her).

The first thing we need to straighten out about this Grand Galactic person is whether in fact he was a he, or indeed a person, or, ultimately, “a” (rather than some fraction of “a”) Grand Galactic.

None of those questions has an easy answer. So what we’re going to do here is, we’re going to ignore the facts and settle for answers that are no problem for us to deal with, apart from their being just plain wrong. First, we will say that this person is really a person, in spite of also being a part of that larger “person” that was all the Grand Galactics combined.

There were Grand Galactics all over the place, from the galaxy’s accelerating fringes to its relatively motionless core, and just about everywhere in between. How many Grand Galactics were there? That’s also a meaningless question. There were many, many of them, but when you came right down to it, the many were also one, because whenever he chose, every Grand Galactic was instantly merged with any or every other.

As you have noticed, we just arbitrarily assigned gender to the Grand Galactics’ pronoun. Don’t assume from this that they practiced sexual intercourse in any sense understandable by a human being. They didn’t. It is just that we can’t go on with that “it” or “he” or “they” business forever, so we just cut the Gordian knot and made him a “he.”

We have just taken one rather large liberty. Let’s take another. Let’s give “him” a name. We’ll call him “Bill.” (Not Bill. “Bill.” It is a major liberty that we have taken, and we should acknowledge that we know it to be so by the use of the quotation marks.)

Now, what else would it be useful for us to know about the Grand Galactics at this time?

For example, would it be helpful to know how big they are? Or at least, since one node of Grand Galactics may be thousands, or billions, of light-years from some particular other node, how they measure distance?

Let’s assume it would indeed be helpful, but we must also recognize that, as with all questions about the Grand Galactics, the answer is going to be complicated. Start with the fact that the Grand Galactics don’t like the kind of arbitrary units of measurement human beings use. When you track those down, they are always based on some human value, such as the distance from a man’s fingertip to his armpit, or some calculated fraction of the distance from a pole to the equator on the particular planet that humans chance to occupy. Grand Galactic measurements are always made on the Planck scale, which is actually quite tiny. The measure of a single Planck unit is 1.616 × 10 -35meters. The easiest way to understand how little that is is to remember that it’s so small it is impossible to measure anything smaller.

(Why impossible? Because you can’t measure anything you can’t see, and nothing can be seen without employing those light-carrying particles called photons. And any photon that was powerful enough to illuminate a Planck-scale distance would be so extremely powerful—and thus so extremely massive—that it would immediately transform itself into a black hole. The word “impossible” is sometimes taken as a challenge. In this case, though, it’s just a fact.)

So to measure anything in any of the three dimensions, whether it’s the circumference of an electron or the diameter of the universe itself, Grand Galactics simply count the number of Planck distances along a line from point A to point B.

That is invariably a large number, but that’s all right with the Grand Galactics. Looked at in one way, they are pretty large numbers themselves.

So, having found ways of at least identifying the un-understandable, let’s get back to that much simpler being, Ranjit Subramanian.

When Ranjit was quite young, his highly ecumenical father encouraged him to read some rather strange books, one of which, by a writer named James Branch Cabell, was about the nature of writing and writers. (For a time Ganesh Subramanian thought that might be a career choice for his son.) What some would-be writers were trying to say to the world, Cabell wrote, was, “I am pregnant with words, and I must have lexicological parturition or I die.”

And, curiously, that is almost exactly the condition Ranjit now felt himself to be in.

For days now Ranjit had been pleading for help, shouting into the empty hallways, explaining, though no one seemed to be listening, that he had something that absolutely had to be communicated to a journal at once. There were no answers. Even the limping old man was now just putting Ranjit’s meals inside the door and limping away as fast as he could.