The table to which Ramanujan referred was a long, dry list of numbers, the product of laborious hand-calculation, that MacMahon had prepared two years before; these were the values, all the way up to p(200), that Ramanujan and Hardy had used as benchmarks against which to check their general formula for p(n). But then Ramanujan had looked deep into MacMahon’s sterile list and made one of those leaps of the imagination that would astonish mathematicians over the years.
To say what he saw, Ramanujan used the language of congruences, which expresses facts about divisibility. Two numbers are congruent when they can be divided by the same number, and leave the same remainder (which may be 0). Take, for example, this straightforward case of division:
14/7 = 2
In the special language of congruences, you say
14 ≡ 0 (mod 7),
which means that you can divide 14 by 7, the “modulus,” with zero left over. And
15 ≡ 1 (mod 7)
means that 15, divided by the modulus 7, leaves a remainder of 1. The number 22 is congruent to 15 because it, too, when divided by 7 leaves 1.
“Congruences are of great practical importance in everyday life,” Hardy with coauthor E. M. Wright would point out in An Introduction to the Theory of Numbers. “For example, ‘today is Saturday’ is a congruence property (mod 7) of the number of days which have passed since some fixed date… . Lecture lists or railway guides are tables of congruences; in the lecture list the relevant moduli are 365, 7, and 24.”
What Ramanujan had found in MacMahon’s table were certain persistent and intriguing patterns best expressed in this simple language. For example, he found that
p(5m + 4) ≡ 0 (mod 5)
Make m anything you like and, whatever you picked, the number of partitions would, Ramanujan showed, always be exactly divisible by 5. For example, let m = 0. Then 5m + 4 is just equal to 4. How many partitions are there of 4? The answer is 5. Is 5 exactly divisible by 5? Why, yes. You could make m = 1,000,000 and ask what p(5,000,004) was and, with not a clue as to what this astronomical number might be, you could say with absolute confidence that it could be divided by 5. Ramanujan also came up with a similar identity that said p(7m + 5) was divisible by 7. Now, in his Cambridge Philosophical Society paper, he proved these results, and conjectured others, one of which would succumb to proof later that year.
• • •
On the same day it got Ramanujan’s congruences paper, the Philosophical Society received another from him, one that brought Ramanujan’s work in England full circle and testified to both the genius and lost potential in his mathematical life.
This second paper went back to two striking identities he had discovered sometime before 1913 and later shown to Hardy. (An identity is an equation true for all values of the variable. So that whereas x − 2 = 3 is an ordinary equation, true only for x = 5, (x − 2) (x + 2) = x2 − 4 is true for every value of x.) One of them was
The other took a similar form. And sometime after Ramanujan’s arrival in England, probably in 1915, MacMahon saw in them something Ramanujan had not—something that bore, once again, on partitions.
Students of the additive theory of numbers, which is the formal name for the field, took an interest not only in partitions generally but in certain classes of them. Take the number 10. The number of its partitions—or to invoke a precision that now becomes necessary, the number of its “unrestricted” partitions—is 42. This number includes, for example,
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10
and
1 + 1 + 1 + 1 + 2 + 1 + 2 + 2 = 10
But, one might ask, what if you excluded partitions such as these by imposing a new requirement, that the smallest difference between numbers making up the partition always be at least 2? For example,
8 + 2 = 10
and
6 + 3 + 1 = 10
would both qualify, as do four others, making for a total of six. All the other thirty-six partitions of 10 contain at least one pair of numbers separated by less than 2 and are thus ineligible.
That’s one class of partitions. Here’s another, formed by a second, quite distinct exclusionary tactic: What if you only allowed partitions satisfying a specific algebraic form? For example, what if you restricted them to those comprising parts taking the form of either 5m + 1 or 5m + 4 (where m is a positive integer)? If you do that, the partition
6 + 3 + 1 = 10
fails to qualify. Why? Because not all the parts, the individual numbers making up the partition, satisfy the condition. The part 6 does; it can be viewed as 5m + 1 with m = 1. So does 1, which can be viewed as 5m + 1 with m = 0. But what about 3? Make m anything you want and you can’t get a 3 out of either 5m + 1 or 5m + 4 (which together can generate only numbers whose final digits are 1, 4, 6, or 9).
One partition that would qualify is
6 + 4 = 10
Another is
4 + 1 + 1 + 1 + 1 + 1 + 1 = 10
Each satisfies the algebraic requirement. In all, qualifying partitions come to six.
Now six also happens to be the number of partitions that fit the first category. Except that it doesn’t “happen to be.” It always turns out that way. Pick any number. Add up all its partitions satisfying the “minimal difference of 2” requirement. Then add up all its partitions satisfying the “5m + 1 or 5m + 4” requirement. Compare the numbers. They’re the same, every time.
This is what Ramanujan’s identity, suitably interpreted, revealed: two seemingly distinct mathematical subworlds fused together in a single unifying relationship. In volume 2 of his book Combinatory Analysis, which came out in 1916, MacMahon devoted to it and the other identity a whole chapter, “Ramanujan’s Identities.” He verified it by reams of hand-calculation as far as any reasonable person could, “so that there is practically no reason to doubt its truth; but,” he added, “it has not yet been established” through formal proof.
But it had been, in a paper twenty years before.
One day in 1916 or 1917 Ramanujan was rummaging through the 1894 volume of the Proceedings of the London Mathematical Society when there, near the bottom of page 318, he saw it. It was entitled “Second Memoir on the Expansion of Certain Infinite Products,” and two of those infinite products were just the identities he thought he had discovered. Before MacMahon’s book came out, Hardy had shown Ramanujan’s identities around. Did anyone know proofs for these marvelous theorems? Could anyone furnish any? No one could. Yet here, in black and white, like a specter from the past, was evidence that someone already had.
“I can remember very well his surprise, and the admiration which he expressed” for the older work, Hardy would say of Ramanujan. As for any loss, or even bittersweet ambivalence, that he may have also felt, Hardy said nothing.
The man who anticipated Ramanujan was Leonard James Rogers, a remarkable character if ever there was one. Born in 1862, in Oxford, where his father was an economist, he had not only done well on the Oxford equivalent of the Cambridge mathematical Tripos, but had earned a bachelor of music degree in 1884. He was a fine pianist, an exceptional mimic, liked to affect a broad Yorkshire accent. He knitted. He skated. He gardened. He was a natural linguist. “Surely,” it was said of him,
no position in diplomacy would have been unattainable to one endowed with his easy mastery of languages, his quick intelligence, his sparkling wit, his fine presence, his athletic grace, his courtly charm that no woman could resist. Yet of what the world counts success he achieved practically nothing.
Rogers was, in spirit, a gifted amateur who, despite his abilities, never pursued his mathematical career with the singleminded devotion so necessary, then as now, to establish a big name: “He did things, and did them well, because he liked doing them, but he had nothing of the professional outlook, and his knowledge of other people’s work in mathematics was vague. He had very little ambition or desire for recognition.”
Even in the years before Ramanujan rediscovered him, he wasn’t quite the mathematical nonentity hi
s obituaries presented him as; for years a mathematics professor at what became the University of Leeds, he had quite a few published papers to his credit. Still, he possessed something of a knack for making notable contributions that were promptly forgotten, at least in the short run. Just three years earlier, for example, the Proceedings of the London Mathematical Society had run this correction:
Prof. Rogers, in his paper “On the Quinquisectional Equation” (Proc. London Math Soc., Vol XXXII, pp. 199–207), gives the equation in explicit form. I regret very much that Prof. Rogers’s work, which I must have seen at the time of [its] publication, had entirely slipped from my memory, so that I was led to state that the problem had not been dealt with since Cayley’s paper in Vol. XII.
In the case of what later became known as the Rogers-Ramanujan Identities, Hardy suggested later, the Rogers originals appeared “as corollaries of a series of general theorems, and possibly for this reason, they seem to have escaped notice, in spite of their obvious interest and beauty.” Then, too, the proofs were tortuous; Rogers’s aesthetic sensibilities didn’t extend to the written word.
Still, there they were.
A correspondence ensued. In the book to which he’d put the finishing touches in April 1916, MacMahon had said Ramanujan’s identities were yet unproved. Now, with the falsehood enshrined in black and white, MacMahon wrote Rogers, as Rogers recalled a little later, “regretting that he had overlooked my work before it was too late.” In October 1917, Rogers wrote MacMahon with a new, simpler proof. In April 1918, probably while at Matlock, Ramanujan wrote Hardy with his own.
On October 28, 1918, the two proofs, along with Ramanujan’s important congruences paper, were read at the annual general meeting of the Philosophical Society.
• • •
Two weeks later, the war ended.
The Bolshevik Revolution had led to a collapse of the Russian armies facing the Germans. Germany had withdrawn all but a million men from the eastern front, rushed them across central Europe and sent them hurtling west toward Paris. Once again, as in 1914, German armies breached the Marne. Once again they came within sight of Paris. But this time, there was a difference. A million fresh American soldiers had arrived in France. The Germans, exhausted by years of war, rolled back, and soon had to sue for peace. The armistice was declared on November 11, 1918.
Very shortly, in the words of the Cambridge Review, “there was a very creditable pre-war bonfire in the Market Place, fed chiefly with packing cases round which there was dancing; but not all the dancers performed with two sound legs.” By one o’clock that afternoon, some Girton College women, normally sequestered in their own campus on the fringes of town, had joined the afternoon revels. At five that evening, many went to the Thanksgiving Service at King’s College Chapel. “As we stood in the dim candlelight of that wonderful chapel,” one of them wrote, “it seemed as if Earth and Heaven were no longer divided, and as if Time and Eternity were one.”
On November 26, Hardy wrote Francis Dewsbury in Madras with word on Ramanujan. “I think it is now time,” he wrote, “that the question of his temporary return to India and of his future, generally, should be reconsidered.”
CHAPTER EIGHT
“In Somewhat Indifferent Health”
[From 1918 on]
1. “ALL THE WORLD SEEMED YOUNG AGAIN”
Ramanujan seemed better and, Hardy wrote Dewsbury, “on the road to a real recovery.” He had gained almost fifteen pounds. His temperature had steadied. The doctors now favored blood poisoning as the source of his ills, and this, it seemed, had “dried up.”
Was it not time, then, that Ramanujan return to India? The reasons against doing so had disappeared. The sea lanes were safe. He had achieved all he had set out to in England. His Trinity fellowship imposed no residency requirement. He need not stick around while his Royal Society candidacy was up in the air; it no longer was. So, the thinking went, with Ramanujan on the mend why retard his recovery by keeping him in England any longer?
Of course, more was going on behind the scenes. “He has apparently been approached (with a view to return) directly by several friends,” Hardy wrote. “It is possible, I think, that the suggestion has not been made in the most tactful way possible; at any rate, it seems to have turned him rather against the idea of going”; something they said had pushed one of Ramanujan’s numerous buttons. Mindful of Ramanujan’s sensitivities, and of no mind to trespass on them, Hardy advised Dewsbury that “the suggestion would best be made more or less officially and by letter simultaneously to him and to me.” Offer Ramanujan a university position that left him free to do research and occasionally visit England and he, Hardy, would favor his return—and Ramanujan, he felt sure, would, too.
Hardy’s letter bore a reminder that the emotional malaise that had catapulted Ramanujan onto the subway tracks earlier that year was not wholly cured. “He will return to India with a scientific standing and reputation such as no Indian has enjoyed before, and I am confident that India will regard him as the treasure he is. His natural simplicity and modesty has never been affected in the least by success—indeed,” he added, “all that is wanted is to get him to realize that he really is a success.”
Ramanujan still didn’t think he was? In November 1918? With close to twenty major papers appearing in the past four years? With an F.R.S. appended to his name? And a fellowship at Trinity College? All this did not convince him? In public, Ramanujan affected a South Indian brand of aw-shucks modesty. But inside, he still wanted something more. Wanted it from Hardy? Was Hardy, on whom the war had made Ramanujan so dependent, in private more niggardly in his encouragement, more aloof and detached, than he was in public?
In any case, the wheels were being set in motion for Ramanujan’s return to India the following year. But was he going back because he was really better? Or because he was worse, his chances for recovery in England seen as remote? That, at least, is what one of his Indian biographers implied later. “Mr. Ramanujan’s disease had assumed serious proportions by the Christmas of 1918,” wrote P. V. Seshu Iyer, referring to a time only a month removed from Hardy’s letter to Dewsbury, “and caused such grave anxiety to his doctors in England, that, hoping to do him good, they advised him to return to his native home in India.”
Whatever was physically wrong with Ramanujan, his progress, or decline, was glacially slow—making it easy to read slight day-to-day fluctuations in his condition any way you liked. So it may have been no great change in his health to which his return to India was really due but, more simply, the end of the war.
• • •
The weeks and months just after the armistice were a time of endings and beginnings, of vast relief all across the blood-drenched soil of Europe. Streetlights blazed again. On December 7, Britons were restored the right to make cakes and pastries and to smear them with chocolate to their hearts’ content. Rationing continued, as it would for some products into 1920, but a double meat ration was announced for Christmas 1918. On December 9, demobilization began.
During the war, 2162 Cambridge men had been killed, almost 3000 wounded—altogether about a third of all who served. Some 80,000 wounded had come in through the rail station over the four years. But now, within months of war’s end, the university was already returning to something like its prewar proportions.
The Nevasa, the British India Lines ship that had carried Ramanujan to England in 1914, went back to carrying travelers and tourists. She had become a troop ship early in the war. Then, her funnel painted yellow and a broad band painted around her white hull in green, she was made over into a hospital ship, with 660 beds, which for two years bore the sick and wounded between Suez, Basra, and Bombay. Later, a troopship again, she carried American soldiers to France, at least twice having to outrun U-boats. Now, in 1919, she was a passenger ship once more.
Hardy’s friend, the historian G. M. Trevelyan, returned to England “to get on with writing history books again. I had no other ambition in life. The delightfu
l delusion that we had done with total war at least for a generation, perhaps for ever, gave a zest to domestic and personal happiness. I shall never forget the exhilaration of a Cornish holiday with my wife and girl and boy at Easter 1919, when all the world seemed young again, and the sands and rocky headlands rejoiced.”
• • •
About the time of the armistice, Ramanujan had left Fitzroy Square for another nursing home, this one in the suburb of Putney, a few miles southeast of London on the south bank of the Thames.
It was called Colinette House. From the outside it was a big, boxy, undistinguished structure, in plan almost square, part of an early suburban development of freestanding brick houses built in the 1880s; Leonard Woolf had lived in one of them on his return from Ceylon a few years earlier, finding it something of a comedown from the Anglo-Indian luxury to which he had become accustomed. The interiors were more impressive, though, laced as they were with elaborate moldings, stained glass, grandly scaled rooms and handsome winding staircases. During Ramanujan’s time, the eight-bedroom house at 2 Colinette Road had been made into a small nursing home presided over by one Samuel Mandeville Phillips.
Compared to Matlock, Ramanujan was much more accessible here, and Hardy (whose mother had died in Cranleigh a few weeks after the armistice) could visit him more easily; beyond the usual two hours on the train into London, Putney was just a cab ride away. Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, “rather a dull number,” adding that he hoped that wasn’t a bad omen.
“No, Hardy,” said Ramanujan. “It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.”
The Man Who Knew Infinity Page 39
