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By Root 1726 0

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