The Last Theorem

4 FORTY DAYS OF DATA DOWNPOUR

So in the remaining six weeks before the new school year began, Ranjit for the first time in his life found himself very nearly drowning in the cascades of the precise sort of information he most desired.

To begin with there were the journals of number theory. There were two major ones in the English language and one or two apiece in French, German, and even Chinese (but he decided early on not to bother with anything he would need to get translated). And the books—so many books! And all now available to him through the interlibrary loan! Ones that looked interesting, though perhaps not directly relevant to his quest, were those such as Scharlau and Opolka’s From Fermat to Minkowski and Weil’s Basic Number Theory, which according to the reviews was not all that basic, indeed quite advanced even for Ranjit. Less promising, because apparently written for an audience not as informed as Ranjit himself, were Simon Singh’s Fermat’s Enigma and Yves Hellegouarch’s Invitation to the Mathematics of Fermat-Wiles and the book by Cornell, Silverman, and Stevens called Modular Forms and Fermat’s Last Theorem. Well, the list was long, and that was only the books! What about the papers, the hundreds, maybe even the thousands, of papers that had been written on this most famous of mathematical conundrums and published—well, everywhere: in England’s Nature and the American Science, in mathematical journals refereed and respected and circulated around the world, and in mathematical journals issued in obscure universities in places such as Nepal and Chile and the Duchy of Luxembourg, and perhaps hardly respected at all.

Somewhat saddeningly he kept finding little curiosities that he would have liked to share with his father. There was, it seemed, a strong tradition of elements of number theory in Hindu literature as far back as the seventh century A.D. and even earlier—Brahmagupta, Varahamihira, Pingala, and, in the Lilavati of all places, Bhaskara. As well as that seminal Arab figure abu-i-Fath Omar bin Ibrahim Khayyám, best known to those who had ever heard of him at all, a number which had not previously included Ranjit Subramanian, as Omar Khayyám, the author of the long collection of poetic quatrains called The Rubaiyat.

None of this was particularly helpful in Ranjit’s dogged pursuit of Fermat. Even Brahmagupta’s famous theorem meant nothing to him since he did not particularly care that in a certain kind of quadrilateral a certain kind of perpendicular would always bisect its opposite side. However, when Ranjit came across the fourth or fifth mention of Pascal’s triangle and the taking of roots in connection with Khayyám, he sat down and composed an e-mail to his father telling of his discoveries. And then he sat for some time with his finger poised over the send button before he sighed and pushed cancel instead. If Ganesh Subramanian wanted to have a social relation with his son, it was his duty, not his son’s, to make the first move.

Four weeks later Ranjit had read, or read part of, every one of the seventeen books and nearly one hundred and eighty papers in his bibliography. It hadn’t been rewarding. He had hoped for some stray insight that would clarify everything else. That didn’t come. He found himself led up a dozen different blind alleys—over and over, because many of the mathematician authors were following the same paper trails as himself. Five or six times each he was reexamining Wieferich’s relatively prime exponents and Sophie Germain’s work on certain odd primes and Kummer’s theory of ideals and, of course, Euler, and, of course, every other mathematician who had innocently ambled into Fermat’s lethally inviting tar pit and, bellowing in fear and pain like any other trapped dire wolf, mastodon, or saber-toothed cat, had never escaped.

The plan was not working. With less than a week before the new school year began, Ranjit faced the fact that he was trying to work too many angles at once. It was something like the very GSSM syndrome Gamini had warned him against.

So he determined to simplify his attack. Being Ranjit Subramanian, his idea of simplifying was to make a head-on attack on that hated and endlessly long Wiles proof, the one that only a handful of the world’s leading mathematicians dared claim they understood.

He gritted his teeth and began.

The first steps were easy. But then he worked further into Wiles’s ugly chain of reasoning and it began to get—well, not hard, exactly, not for the likes of Ranjit Subramanian, but at least it began to require concentrated attention for every line. Because that was when Wiles began considering the equations for curves in the x-y plane, and for elliptical curves, and for the many solutions to the equation for modularity. Which was when Wiles, for the first time ever, was able to demonstrate that what was called the Taniyama-Shimura-Weil conjecture—namely, that any infinite class of elliptical curves was modular—was valid. And then, while Gerhard Frey and Kenneth Ribet had demonstrated that a certain elliptical curve could not be modular, Wiles himself was able to demonstrate that that same curve necessarily had to be modular….

And, aha! There it was! A veritable contradiction!

A contradiction was the mathematical pot of gold that—sometimes!—lay at the end of some interminable mathematical trail. A contradiction was the thing that mathematicians gladly devoted their lives to finding, because if logical deductions from your starting equation wind up with two conclusions that contradict each other, then your starting equation itself must be wrong!

And so it was proved—sort of proved—that Fermat had spoken truth. The square was the limit. The sum of no two cubes would ever be another cube, and so on for every other exponent this side of infinity. But Ranjit was no nearer to finding his own less daunting proof of what Fermat had so casually mentioned so long ago.

And—oh, yes—he was not aware that his picture was being taken.

The beings that were doing the picture taking were another of those client races of the Grand Galactics. These were called the Machine-Stored, and of course Ranjit never saw them. They didn’t intend to be seen. They generally weren’t ever seen, either, although under certain rare combinations of starlight, moonlight, and gegenschein, a few of them had occasionally been detected by an occasional human being. When reported, these sightings were generally referred to as sightings of flying saucers, thus adding to the vast catalog of fakes, mistakes, and downright lies that made it nearly certain no respectable scientist would ever pay any attention to them.

What the Machine-Stored were doing on Earth at that time was anticipating a need of the Grand Galactics, whose needs and wishes the Machine-Stored always catered to. The Grand Galactics hadn’t ordered this activity, but the Machine-Stored were permitted to use their own discretion in certain limited circumstances. The special quality of the Machine-Stored lay in the fact that they had trashed their planet even more diligently than the One Point Fives, to the point where organic life on its surface was now completely impossible. The One Point Fives had dealt with the problem by adding infinite prostheses to their vulnerable organic bodies. The Machine-Stored took a different tack. They abandoned their physical planet, and indeed abandoned everything physical at all. They reconstituted themselves as something like computer programs and allowed their now quite frail and sickly bodies the privilege of death, while the individuals lived on in cyberspace. (Since which their despoiled planet had begun to show the beginning signs of regeneration. Not all of its liquid water was now toxic, for instance—though it still would have been pretty much a hellhole for anything organic.)

And the Machine-Stored themselves?

Why, they made themselves useful. Sometimes, when the Grand Galactics chose to move a certain quantity of objects or beings from one star system to another, the Machine-Stored were instructed to do the moving. And when they had detected those first microwave and then nuclear pulses from Earth, they’d known the Grand Galactics would take an interest. They didn’t wait for orders. They at once began to survey the planet and everything on it, and to pass those findings at once to the corner of the galaxy where the Grand Galactics swam in their dark energy streams.

Of course, the Machine-Stored had no good idea of just what the human race was up to in its various activities. For that, they would have needed to understand human languages. That didn’t happen. The Grand Galactics preferred that their client races be ignorant of any languages but their own, because if the races could freely talk to one another, who knew what they might be saying?

Ranjit would have been astonished to know that his own picture had been flashed across interstellar space in that manner. It had, though. So had the pictures of everyone, and almost everything, on Earth, because the Machine-Stored—if not omnipotent—were diligent.

And hoped that the Grand Galactics would appreciate, or at least tolerate, that diligence.

When Ranjit’s bedside radio woke him for the first day of his new term, he leaped out of bed in order to turn it off. His first class, Astronomy 101: The Geography of the Solar System, was also pretty close to his last hope that the university would provide him with anything interesting over the next three years. That was mildly cheering in itself. And then, as he was leaving the building, the porter handed him a letter—from London, and therefore from Gamini—and Ranjit actually did feel a little bit cheerful.

Hunched over his breakfast, he read the letter. It didn’t take very much time. The letter was even shorter than its predecessor and almost entirely devoted to describing Gamini’s “superb maisonette”:

Ranjit managed a smile at the thought of Gamini doing laundry—the Gamini for whom laundry had always been what you took home and gave to the servants, who the next morning would return it to you, cleaned, ironed, and folded.

That did not keep him from wondering just who this Madge was.

So he showed up for his first class prepared for disappointment….

But then, wonderfully, miraculously, that was not at all what happened!