Hidden Harmonies

by Ellen Kaplan

  The story is well known about the Oxford mathematician Hardy visiting his extraordinary protégé Ramanujan in the hospital, and remarking that the number of the taxi he had taken, 1729, was rather dull. “No, Hardy,” said the dying Ramanujan, “it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.” But what if Hardy’s cab number had been 13,852,800? Or 211,773,121?

  You remember the rectangular box of Chapter Six, which we traded in for all of space in Chapter Eight. Here it is again, like one of those stubborn gifts of folklore that you just can’t give away. We found that we could climb up out of the plane on its body diagonal, and then up an endless Pythagorean spiral of diagonal steps into n dimensions. Since its flattened version can have all three sides be whole numbers in infinitely many ways (the Pythagorean triples), might there not be at least one way for the seven relevant lines of its three-dimensional version to be whole numbers too—the three edges a, b, and c, the three face diagonals, and the body diagonal?

  In 1719 a German accountant named Paul Halke found a box with all its edges and face diagonals whole numbers: a = 44, b = 117, and c = 240 (giving the face diagonals 125, 244, and 267)—but the body diagonal, , was irrational.

  The blind eighteenth-century English mathematician Nicholas Saunderson, inventor of a calculating machine called his Palpable Arithmetic, who lectured on optics, among other subjects, at Cambridge University (the man he replaced “was dismissed for having too much religion, and Saunderson preferred because he had none”, wrote his friend, the astronomer Halley) came up, in the sixth book of his Elements of Algebra, with a way—now called Saunderson’s Parametrization—for finding many rectangular boxes with whole numbers for all but the body diagonal. Halke’s box is the smallest of these. Unfortunately, in 1972 W. G. Spohn, at Johns Hopkins, proved that no Saunderson box can have an integral body diagonal.

  Our species has not been idle. What about boxes with only one edge irrational? In 1949 “Mahatma” (in a London journal for assistant masters) asked readers for solutions, and received an infinity of them—one of which (oh Hardy!) involves the numbers 13,852,800 and 211,773,121.

  Think back to those mistaken readings of Plimpton 322 by Neugebauer, which included values for Pythagorean triples like 12,709. Why does this Theorem call up such gigantic figures from Nightmare Land? Has every number the mark of personality on its brow? If not each separate number, what never fails to startle is the presence among them of subtle structure: many of Mahatma’s sextuples form cycles of four—and if we turn to sextuples with only a face diagonal irrational, solutions occur in cycles of five!

  But a perfect box? No Mahatma nor any palpable arithmetic has yet found one, nor has yet shown that none can exist. Informed opinion has it that “for the moment, this problem still seems out of reach.”an

  THE FERMAT VARIATIONS

  You may have wondered why, of all the generalizations we’ve explored, no mention has yet been made of the most famous: changing the exponents in x2 + y2 = z2 from 2 to higher powers. Fermat’s centuries-old boast that there are no non-zero integral solutions x, y, and z for any exponent beyond n = 2 was answered in 1995 (how oddly dates sit among timeless truths) by Andrew Wiles and Richard Taylor, who proved that Fermat was right. Long before, Euler may have been the first to show no solutions exist in the particular case n = 3.

  But what about possible solutions for x3 + y3 = dz3, where d is some natural number other than 1? Or for other n > 2: x n+ y n= dzn, or for mixed exponents in this format? Too much of a muchness: since the case n = 3 has proven difficult enough (we have only the most tantalizing clues), we’ll now look just at the state of play for it.

  We know, for example, that of the infinitely many possible values for d, when d = 7, what works is x = 2, y = −1, and z = 1:

  There is a growing anthology of other results, with some large entries: if d = 967,

  and—oh yes, some very much larger, as one with an x having 51 digits (when d = 2677).ao

  For x3 + y3 = dz3, we know some d for which there are only a finite number of solutions, like d = 3. Others (d = 13, for example, or 16) splendidly produce an infinite number. Finding such d isn’t actually hard, but it isn’t the real issue; finding the structure of the solutions for a specific d is. Think of the complexity of the solutions we know so well for x2 + y2 = z2, with its fan of practical applications and theoretical implications. Will there be similar riches for each of the d in our cubic equation? Will each star of a d in this x3 + y3 = dz3 nebula have its own idiosyncratic planetary system? What is the way in from this question—and is it a way into a jungle or an overgrown city of observatories?

  Let’s narrow down our looking a second time to a more manageable vista, and work from now on with some specific d. You’ll find that we stand at the entrance to an invitingly green tunnel.

  You remember from Chapter Seven that the modern way to find all whole numbers x, y, z satisfying x2 + y2 = z2 was to recast them as rational points on a circle of radius 1, and then let a secant line pinned on this circle at (–1, 0) sweep through it, reading off intersections with the circle when its slope was rational (for then these intersections would have rational coordinates too). The hidden power of this approach is that it generalizes from circles to other curves on the plane—such as our x3 + y3 = dz3, when it too is rewritten as .

  Once again we would let a straight line anchored on it, at some point with rational coordinates, sweep its contours, arresting it when the line’s slope is rational. Unfortunately, in this new context, that won’t guarantee that the new point of intersection will also be “good”—i.e., have rational coordinates. It would, however, had we drawn our line through two good points, for then the line itself will have rational coordinates, and will intersect the curve in a third good point; this leads to more secant lines and more good intersections, in a nicely reciprocal way, either forever, or until the process eventually closes up in a finite circuit.

  The point-and-secant technique on a generic cubic curve.

  What is the minimal number of these ‘starter points’ needed, for a particular d, to generate all the rest? Thanks to Louis Mordell—nicknamed ‘X, Y, Z’ by his schoolmates—we’ve known, since 1922, that this minimal number will always be finite—but will it be large for some d and small for others? Will they even all lie below a uniform upper bound? We don’t know, we don’t know.1

  What’s alluring in all of this ignorance are the merest whiffs of intriguing structure drifting through, here and there, suggesting patterns of profound order: gold threads woven into a cushion tucked in the howdah on the back of the elephant that many a mathematician is groping over in the dark.

  THE ABC CONJECTURE

  What a universe there is of equations with three variables a, b, and c, standing for integers, and raised in different ways to powers. Call this universe Diophantine, after that astonishing explorer, Diophantus, whom you met briefly in Chapter Seven.

  We know how to find the infinitely many solutions of a n+ b n= c n when n = 2, and Fermat tells us that there are no solutions in natural numbers when n > 2.

  The later Fermat-Catalan Conjecture claims that only finitely many solutions exist for a m+ b n= c k, (where all of these letters stand for natural numbers greater than 1; a, b, and c are relatively prime, and m, n, and k obey the further constriction that (1/m) + (1/n) + (1/k) < 1). Ten solutions—such as 25 + 72 = 34—have been found so far, but the conjecture has yet to be proved.

  The Dutch number theorist Robert Tijdeman suggested in the 1970s that there are also only finitely many solutions for am+ d = bn, for a fixed integer d greater than 1, with a, b, m, and n integers greater than 1. This too remains unproven.

  And a page or two ago we flew past the spiral nebula a3 + b3 = dc3, seeing many solutions when d = 7, say, and knowing there are infinitely many when d = 13 or 16, for example.

  Mostly, however, this Diophantine universe is a mystery to us. If we can’t yet tell its shape and
the forces holding it together, could we at least find out in general how to distinguish cases with finitely many solutions from those with infinitely many? Or at any rate come up with a sufficient condition for the former—a condition that would say: if such and such is true, then there will be only a finite number of solutions?

  Such a condition was proposed in 1985 by Joseph Osterlé and David Masser: the ABC Conjecture. To put it in a slightly weakened form: there will be only finitely many solutions when a + b = c, all are relatively prime positive integers, and c is greater than r2, where r is the product of the primes dividing a, b, and c. Even in this weakened form, Osterlé and Masser’s conjecture is at present out of our reach.

  It took 358 years to settle Fermat’s Last Theorem, but it has no hinterland: no structural consequences. Why should the ABC Conjecture seem to lead, past the proofs of these conjectures, to a deep understanding of our Diophantine universe?

  You wouldn’t be surprised were such a dynamo almost impenetrably packaged, like your car’s control module—or, for that matter, like the brain in your skull. To grasp what effects the conjecture would have, were it proven, would be to bring yourself up-to-date in modern number theory.

  And as for whether there really is a mechanism inside the housing, and it works, and its gears can be meshed with those outside it in a proof—that seems a prospect more like decades than years away. As you read this, a grid computing system in Holland is churning out triples (a, b, c) with c > r2, as its programmers scan the output for patterns; and people around the world are teasing at and tinkering with the machinery that drives this condition—which is anyway only sufficient, not necessary, to distinguish merely between finitely and infinitely many solutions to the cluster of problems in this universe. We aren’t even yet, it seems, in the antechamber to the presence room outside the hall where the profound answers are enthroned.

  OUTWARD FROM SAMOS

  Oh, and how many right triangles have rational sides and integral area? And can we find all non-Euclidean Pythagorean triples? And . . .

  When does no more than a puzzle become a profundity, with the pieces all locked together in this Chinese box of a world? When it becomes structural.

  And is Pythagoras not endlessly reborn past the thousand proofs, in the generalizations and conjectures of what passes for his theorem?

  The world as it yet might be, unguessed, cradles the world as it only is.

  AFTERWORD

  Reaching Through—

  or Past—History?

  “I often think it odd,” says Catherine Morland in Northanger Abbey, “that history should be so dull, for a great deal of it must be invention.” The philosopher J. L. Austin (who always hoped his books would be mistaken for Jane’s) couldn’t have sparked a livelier debate. What should history be—objective or subjective? Could it be either, or both? These are especially sharp alternatives in our little corner of it, where we have so much to do with proofs—which are ligatures in time to eternal truths. Yet the proofs are the work of provers, and the words of the leading Pythagorean scholar, Walter Burkert, are a tinnitus in the ear: “One is tempted to say that there is not a single detail in the life of Pythagoras that stands uncontradicted.”1

  Perhaps there are no facts, as a sparring partner of Austin’s once claimed. Ours, at least, are very far away. Theory crusts them over. Intention generates names and dates, and intentions are puff ball.

  Looking back, we lose sight too of those minute, informing leanings, so negligently known at the time, that ended up prodding balanced alternatives in the direction they ultimately took. We replace them, in our architectural zeal, with abstract forces grandly materializing in events: the Being that brackets all Becoming. This Being, however, has noticeably contingent qualities. We still cast it largely in a linear, causal mode rather than that of a network. You never hear anyone ask, “What is a meaning of things?”

  In the Freer Gallery of Art in Washington there hangs a magnificent Nocturne of Whistler’s, showing Battersea Reach. There is the bridge, and there a barge beneath. Those ripples of yellow are riding lights; the darker verticals are masts and, beyond, stacks on the bank. Or it is a quartering of the canvas into water, fog, and moonlight—or rather, into lighter blues shading to darker: not so much representation as composition? Expert knowledge of time, place, and biography might reveal further details of the shore—or the brushwork—or the intentions. Past a certain threshold of analysis, however, we would come up with only ground ultramarine and canvas threads. To press too far beyond an intriguing scatter of data is to study not history but historiography.

  Historians need an inner justicer to arbitrate among their capacities: an eye for details, an imagination to vivify, and a mind to deduce the likely to and from them, with an engaged spirit relishing presence and seeking out significance. We easily get lost in these contrary enterprises, taking our singular selves as the measure of all things, turning into unwitting partisans of those we study, recasting a simplifying hypothesis as world order. And there are all the little treasons of history’s clerics, like reading the prior generation’s orthodoxies as heresy. This was once thought Oedipal but is no more than opportunistic, in the pursuit not of history but of the historian’s profession.

  With our scanty evidence so corrupt and corruptible, what should we do? In what tense should our stories be written—the historical past, whose every ending acknowledges its fictional status? In what mood—the subjunctive, where “must” is no more than projected “ought”? Or aim to establish as probable a context as we can shore up in a sea of the possible, so that within it we may measure in common with distant others the depth of a startled insight, for all the incommensurability of our lives?

  Yet there is the outlook above, starring each of our skies with theorems and their awesome premonitions of order. Let us indeed eke out the lives of the provers with our imaginations, and reckon the past from the bearings this inner compass gives: but our course is set by the un-canniness of impersonal truths. “He whose oracle is in Delphi,” said Heraclitus, “neither affirms nor denies, but indicates.” And Pythagoras, bending over the tripod of the theorem that bears his name, no more than points toward the abstract, which informs our every singular.

  Acknowledgments

  We’ve been very lucky in the inspiring help we’ve had, with so many different aspects of this book, from Barry Mazur, Amanda and Dean Serenevy, John Stillwell, Jon Tannenhauser, and Jim Tanton. Eric Simonoff has been, as ever, a stimulating and benign presence. The OED speaks of an editor as one who gives to the world. Peter Ginna exemplifies this: the book was his idea, and his patience, humor, and encouragement have brought it out.

  Our immense thanks to friends, acquaintances, and strangers who have variously set us straight or pointed out a path where we saw none: David Domotor, Grant Franks, Gene Golovchenko, Jean Jaworski, Ulla Kasten, Oliver Knill, Jim Propp, Jan Seymour-Ford, and Michael Zaletel.

  Notes

  CHAPTER 1

  1. In Aldous Huxley, Little Mexican (London: Chatto & Windus, 1924).

  2. Heath, A History of Greek Mathematics I.142. Heath notes the anachronism: Thales precedes Pythagoras. On the other hand, it isn’t Pythagoras whom Callimachus names but the Homeric hero Euphorbus, whom Pythagoras claimed to have reincarnated, and who did indeed precede Thales.

  3. Ibid., I.121, citing Herodotus II.109.

  4. Ibid. Heath points out that this claim occurs in Greek historians, who may all be no more than elaborating on the bare statement, given above, from Herodotus.

  5. Wikipedia, “Berlin Papyrus”, http://en.wikipedia.org/wiki/Berlin_Papyrus.

  6. The passage on the stone circle in Egypt is from http://www.philipcoppens.com/carnac.html, “Counting Stones”. On the analysis of Thom’s data, see M. Beech, “Megalithic Triangles”, Journal of Recreational Mathematics 20, no. 3 (1988).

  CHAPTER 2

  1. The passage is from Samuel Noah Kramer, “Schooldays: A Sumerian Composition Relating to the Educa
tion of a Scribe”, Journal of the American Oriental Society 69, no. 4 (October– December 1949): 208.

  2. The outline of our summary is based largely on Høyrup, “Mesopotamian Mathematics”.

  3. Friberg, A Remarkable Collection, 434fn.

  4. Robson, “Words and Pictures”, 111–112, and her “Three Old Babylonian Methods”, 70.

  5. Ramanujan to Hardy, letter of February 27, 1913: “If I tell you this [namely, that the sum of all the natural numbers equals −1/12], you will at once point out to me the lunatic asylum as my goal.”

  6. Friberg, Amazing Traces, 36.

  7. Herodotus II.109, Diogenes Laertius II.1–2, Suda, s.v., cited by G. S. Kirk, J. E. Raven, and M. Schofield, 100–103.

  8. Aristotle, Physics G4, 203a10.

  9. As listed in Friberg, Remarkable Collection, 449: “Damerow listed all mathematical cuneiform texts known to him, in which the theorem is used either directly or indirectly. . . . Høyrup wrote a brief note with a discussion of nine [further] examples. . . . The Høyrup/Damerow list is updated below, and made considerably more complete and explicit.” We have confined ourselves to Old Babylonian texts only, since examples from the Late Babylonian/Seleucid might well have been back-influenced by developments in Greece.

  10. Robson, “Neither Sherlock Holmes nor Babylon”, 185.

  11. Jens Høyrup, “Changing Trends”, 23–24fn.

  12. Robson, “Neither Sherlock Holmes”, 185.

  13. Høyrup, “Changing Trends”, 24.

  14. David E. Joyce, “Plimpton 322”, http://aleph0.clarku.edu/~djoyce/mathhist/plimpnote.html (1995).

  15. Robson, “Neither Sherlock Holmes”, 182–83; Jens Høyrup, “Mesopotamian Mathematics”, in Cambridge History of Science (Cambridge: Cambridge University Press) I.7.

  16. R. Creighton Buck, “Sherlock Holmes in Babylon”, American Mathematical Monthly, May 1980, 335–45.