# Tres Nombres

Édouard Lucas (1842-1891), a French mathematical prodigy, died in a horrifyingly unusual, unexpected way.

He was 49, a professor in Paris, the sweet beating heart of mathematics at the time; he had just published his *Théorie des nombres*, a magisterial exposition and exploration of number theory, and he had declared to the world his work for a second part of the same --- and to those he had thought his friends, his plans for a third part, which would contain metaphysical speculations arising from the results of the preceding two.

So in 1891 Édouard Lucas was a king of Paris, a revealer of such vistas as the yokelous masses of minor mathematical dabblers had never seen; a man that had bargained with the sweet numinous numeriform Goddess of Sciences herself, and wrested from her bosom theorems of glittering profundity and staggering augurity; and he was the guest of honor at a meeting of the *Association française pour l'avancement des sciences* --- when he was savagely, fatally, assaulted.

A waiter --- whose face no-one present saw, whose escape no hand nor word thought to obstruct --- came up behind Lucas, and dropped a tray full of heavy crockery on his head. Lucas's skull was crushed, he was rendered unconscious without the opportunity to say a single word; and though the crushing itself was not fatal, septicemia killed him a few days later. He never regained consciousness.

His papers, the plans for the final two volumes of *Théorie des nombres* among them, passed to the keeping of young André Gérardin, a minor mathematician. There were many that would have dearly liked to see those papers; but Gérardin, their keeper, would not allow this. He became increasingly eccentric with his obstinacy as the years passed. Sometimes he would let a visitor in, but in into his hovelous one-room apartment just to gaze on a rough wooden box; the container of Lucas's jewels, he would gloat, and whether the visitor was or was not satisfied with this external view of planks and nails, he was soon driven out by Gérardin, muttering curses, threats and demands for obscene rewards.

In 1953 Gérardin died, penniless, more than half-mad and pitiful. Mathematicians gathered like vultures to his funeral, whispering of the box --- the box of Lucas --- the box of secrets.

That maddening container was pried open --- and what was found inside were merely copies of Édouard Lucas's published papers. His notes, his drafts, his plans, his hopes and dreams were lost; had been lost for over half a century, and were now beyond all tracking and recovery. The second volume of *Théorie des nombres* passed into the realm of dashed hopes and muttered curses --- "Laisant said Lucas had shown him a proof of Fermat's Last Theorem! To think of such a result, lost!" --- and the third, the volume of metaphysical verities and speculations, into legend, myth and finally into oblivion, as a pursuit not willingly remembered in connection to such a giant of reason and intellect as Édouard Lucas, the prime of Paris.

It is not known if Gérardin destroyed the notes, or lost them some other way. Perhaps they were stolen from him, or lost in some shameful swindle or gamble, and the poor man never could admit it. Thus he was reduced into the madness of taunting the curious with a box that was all he had, a box that he alone knew did not contain what the curious sought.

*

Fermat's Last Theorem, probably the most famous puzzle in the theory of numbers, was proven by Andrew Wiles in 1995, mere 358 years after it was formulated.

The puzzle was, admittedly, neither Fermat's last nor a theorem: a conjecture, a mathematical hypothesis, can be called a proper theorem, a proven result, only after it had been demonstrably proven.

This result gained its notoriety from the fact that Fermat had written it down, noting the proof was "marvelous", but too long for that spot on the book's margin. It is generally believed that though the result is true (demonstrably, after Wiles), there could not have been any proof so obvious and marvelous as to fit what Fermat's note had implied.

The actual statement of the theorem is both dry and easy to understand, even for a layman: it is that if we denote with a, b, c and n any four integers, and require n to be greater than two, then we cannot choose the four so that a^{n}+ b^{n} equals c^{n}. (What we mean by a^{n} is the number a multiplied with itself n times: a^{3} or "a cubed" is a times a times a.)

This result does not at first glance seem to be of earth-shattering importance, and any exposition of its associations, history and proof likewise do not make it appear more than a curiosity, one whose fame lies mostly in that damnable "truly marvelous proof" of Fermat's careless pen.

As not even mathematicians are immune to the lure of such populist fascination for the sake of longievous mystery, there had been many prizes promised for the proof of the Last Theorem; and consequently, the theorem also enjoyed a reputation as the (assumed) result with the greatest number of blundering, ignorant, foolish attempts at proving it. During the worst years, one particular prize-promising instance endured a staggering 621 mailed attempts in a year, most from (and I quote) "sad people with a technical education but a failed career --- people that possessed the desire, that were possessed by the desire, that were daemoniacs ridden by the riddle of the Last Theorem --- but that did not have the education to see how wrongheaded their attempts were."

Now, why such a long account of a problem that has been solved --- a conjecture that has become a theorem, already?

For this singular reason: Though Andrew Wiles proved the theorem, his proof was singularly long, baroque and involved; centuries beyond the grasp of Pierre de Fermat and his careless ink, quill and mind. The minds of mathematicians are still haunted by this ghost: maybe there's a shorter way, a much shorter way, a proof that is sweet, elementary and obvious; a proof that does not require the invention and linking of entire new fields of mathematics to solve this one problem. To use 100 pages for the proof of a statement that does not take as many letters seems monstrous; to bloat it into a delirious rampage through elliptic curves and Galois representations and Hecke rings seems, to many mathematicians, a grotesque mockery of how things ought to be done, in a beautiful world.

Maybe in this one case the beautiful world of mathematics is not beautiful; maybe it has this crevasse of gibbering lunacy, a bottomless dark pit where congruent demons and perverse imps of cohomology drag the unwary down to endless flights in lands distant, distasteful and unknown; maybe it is so. It is very possible what Wiles found is the best route, if not outright the best way that route could take. But still, mathematicians hope for a better, simpler route: a proof of ten lines, not one hundred pages.

And this proof was in the hands of Édouard Lucas, if the statement of Laisant is to be believed; the statement of Laisant, a character of as much honesty and nobility as only a Parisian mathematician of the turn of the century could be. That proof was in the drafts for the second volume of *Théorie des nombres*, a work to be even more profound than the first. That proof was, in essence, not Wiles's maze of serpentine arguments and traps for the ignorant, but rather a ladder, a stairway to the high aether of mathematical understanding: to the vistas that had given Édouard Lucas ideas for the proposed, rumored third volume of his work, the metaphysical speculations of matters and manners within and beyond all time and space.

But those notes are lost; Gérardin lost them, or they were taken from him; and their keepers do not reveal their secrets even as much as Gérardin did.

But still, is that such a loss? Maybe Lucas was as mistaken as Fermat was, and his proof was fatally flawed. Maybe Laisant misremembers. Maybe Gérardin's madness was not because of loss, but for disappointment: the disappointment of peering into the notes of a giant, and finding only the work of giant in winter, spent of energy and invention. Why should we care? Why does a short proof for that curiosity of the Last Theorem, or any other of Lucas's rumored results, *matter?*

To this, I wish I could answer with a shrug, with a glib dismissal, with an admission they matter not, not at all, that they are one more fascination for fascination's own sake, without any further utility --- but I know better.

It is the constant tragedy of mathematics, as naively viewed, that though its practitioners do not desire a worldly life, do not desire the moral troubles of nuclear physics and Frankenfood biology, they cannot ever invent anything so pure and ethereal that it should not be, sooner or later, put into applied and often abhorrent use. What would artillery have been without the trigonometry of mathematicians' abstract fancies? What would the atomic bomb have been, without the mathematics used to foresee it?

What would your computer, your cell phone, be without the mathematical secrets that keep your secrets have? Cryptography, the concealment of written secrets, is a most eminent practical field --- but without mathematics it would not exist. And the most useful field of mathematics for its robed cavorting is number theory: one of whose foundations is the uncertain hulk of Fermat's Last Theorem.

Entire libraries of number theory were written *around* the Last Theorem: it was widely assumed to be true, thought to be true, believed to be true: but a mathematician will not believe until he sees, will not accept what the hand of his mind cannot grasp and hold. For this reason mathematicians have ever been the most vociferous atheists, and believers. If the Last Theorem could not be shown true, it could not be used to build further results: and so instead of the straight way, a bent path was made into the dizzying aethers of numerical mysteries. Wiles's proof changed little of this: the short way was open, but its walls held such frightful alien murals few wanted to move from the old familiar crooked path.

But imagine if there was a simple, short proof of the same. That would have the earth-shattering profundity and impact of a door in London that opened into the arid deserts of Mars's Valles Marineris! And just as such a door would, after a moment's hesitation, be conduit to howling winds liable to lay both planets to waste, so would a short proof of Fermat's Last Theorem cause a tumult of insight and inspiration that would reduce a host of number theory's most puzzling problems into insignificance: and in their immediate application, destroy every cipher, cryptography and code ever devised.

Whosoever possessed that proof and a shred of mathematical ability, and a touch of black ambition, would in due time find all the secrets of mankind opened to him. And Gérardin lost the papers, or they were stolen from him!

It is with despair that I read of attempts in simplifying Wiles's proof, for I know the work is already done; the truth, the simple proof is already out there, and has been for a century. There are unscrupulous men that do not desire attention or raw earthly pleasures; men bent and crooked in worship of a cold, callous Goddess of Sciences, men (and occasionally women) of secretive, obscure way, obsessed with knowledge and certainty. I speak of a certain cabal among mathematicians, a cabal whose members I will not name for I do not wish to be ruined; but they have the result, and as they possess it, there is no point in trying to conceal our words and opinions. They have the Universal Key, the solution to all cryptography, and all codes are as plain as the purest, most flawless glass to their malign glare.

He who has eyes, let him see.