Ramanujan Sent 120 Theorems With No Proofs, So Hardy Decided They Must Be True

In 1913 a self-taught clerk in Madras mailed more than a hundred proofless theorems to Cambridge. Hardy saw genius in them, brought the clerk to England, and for six years the two reshaped number theory. Then illness, a death at thirty-two, and a notebook no one opened for fifty-six years.

The letter on the breakfast table

On a January morning in 1913, the Cambridge mathematician G. H. Hardy sat down to breakfast at Trinity College and found, among his post, an untidy envelope with Indian stamps. Inside were ten pages of mathematics and a covering letter from a man he had never heard of. The theorems were wild, most stated without a word of proof.

Hardy's first instinct was to file it with the crank mail every prominent mathematician receives. His friend C. P. Snow, recounting the story Hardy told him, described the morning this way: Hardy "gave the manuscript a perfunctory glance, and went on reading the morning paper." But the formulas nagged at him through a lecture and a game of tennis, and the question forming in his mind was sharp: is a fraud of genius more probable than an unknown mathematician of genius? That evening he took the pages to his collaborator J. E. Littlewood, and by midnight they had their answer. The writer was a man of genius.

The letter, dated 16 January 1913, was from one S. Ramanujan, a clerk in the Accounts Department of the Port Trust Office at Madras, on a salary of £20 a year. He introduced himself plainly, as the published record preserves it:

"I beg to introduce myself to you as a clerk in the Accounts Department of the Port Trust Office at Madras on a salary of only £20 per annum. I am now about 23 years of age. I have had no University education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at Mathematics. I have not trodden through the conventional regular course which is followed in a University course, but I am striking out a new path for myself."

He had, he wrote, made "a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'." He claimed to have found a formula, "the error being negligible," for the number of primes below a given number, a problem Hardy's own tract Orders of Infinity said was unsolved. Then the pitch: "Being poor, if you are convinced that there is anything of value I would like to have my theorems published... Being inexperienced I would very highly value any advice you give me." The enclosed pages held more than a hundred theorems.

What made Hardy take it seriously was not the familiar results, nor the ones he could prove with effort, but a handful of formulas on a different level, the ones he later singled out in his book Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work (1940): they "defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them." His test was the impossibility of forgery: no fraud could have dreamed these up.

Who was this clerk? Srinivasa Ramanujan had been born in Erode, in what is now Tamil Nadu, on 22 December 1887, and raised in the temple town of Kumbakonam, where his father kept a cloth merchant's books. He had almost no competent instruction. At sixteen he borrowed G. S. Carr's Synopsis of Elementary Results in Pure and Applied Mathematics, a cramped Victorian cram-book that listed thousands of theorems with only sketchy proofs. It became his model of what mathematics looks like, a list of true results, and it is one reason his own work arrived for years as conclusions without derivations. He won a scholarship to Government College in Kumbakonam, then lost it for neglecting every subject except mathematics. At Pachaiyappa's College in Madras he passed mathematics and failed everything else, twice failing the First Arts examination, which closed the door to the University of Madras. By his early twenties he was poor, unemployable, and filling notebooks with identities on infinite series, continued fractions, and primes that no one in Madras could evaluate.

His friends in the small Indian Mathematical Society kept him afloat. Ramachandra Rao, the society's founder, gave him a stipend for a time. When that ran out, Ramanujan found work as a clerk in the Madras Port Trust, where the manager, S. Narayana Iyer, and the chairman, Sir Francis Spring, let him work mathematics at his desk. It was from this post that he wrote to Hardy, having first written, unsuccessfully, to two other English mathematicians. Only Hardy answered.

In his second letter, of 27 February 1913, Ramanujan dropped the formality. "I am already a half-starving man," he wrote. "To preserve my brains I want food and this is now my first consideration. Any sympathetic letter from you will be helpful to me here to get a scholarship." Hardy, meanwhile, had replied on 8 February with a professional's candor: he was "exceedingly interested," but he needed proofs. The results, he said, fell into three classes, some already known, some new but curious, and some "which appear to be new and important." The third class was the one that mattered. With Hardy's backing and a word from a former Trinity man, Gilbert Walker, the University of Madras broke its own rules to grant Ramanujan a special research scholarship of Rs. 75 a month. For the first time, he was paid to do mathematics.

Six years, and the quarrel over proof

Hardy wanted Ramanujan in Cambridge. Ramanujan, an orthodox Brahmin, hesitated, fearing he would lose his caste by crossing the sea, until E. H. Neville, a Trinity colleague of Hardy's lecturing in Madras, persuaded him and his family. He sailed on 17 March 1914 and reached Cambridge in April, staying nearly five years.

The collaboration that followed is one of the strangest in science, because the two men worked in opposite ways. Hardy was a rigorist for whom a proof was the whole point; Ramanujan worked by intuition, arriving at results he often could not explain. Where the formulas came from was, to him, partly a religious matter. His family goddess was Namagiri of Namakkal, and his Indian biographers, Seshu Aiyar and Ramachandra Rao, wrote that "Ramanujan used to say that the goddess of Namakkal inspired him with the formulae in dreams... he would note down results and rapidly verify them, though he was not always able to supply a rigorous proof." To a friend, S. R. Ranganathan, who published the recollection in his 1967 memoir Ramanujan, the Man and the Mathematician, he once said, turning mid-explanation: "Sir, an equation has no meaning for me unless it expresses a thought of GOD."

Hardy, an atheist, said so bluntly in print. "I am sure that Ramanujan was no mystic," he wrote, "and that religion, except in a strictly material sense, played no important part in his life." He recorded Ramanujan telling him that "all religions seemed to him more or less equally true." The two memoirs, Hardy's and the Indian biographers', "contradict one another flatly about his religion," Hardy observed. The reader is left holding two accounts that cannot both be whole. What is not in dispute is the work.

"The limitations of his knowledge were as startling as its profundity," Hardy wrote. "Here was a man who could work out modular equations, and theorems of complex multiplication, to orders unheard of... whose mastery of continued fractions was beyond that of any mathematician in the world." Teaching him was delicate: "It was impossible to ask such a man to submit to systematic instruction... So I had to try to teach him, and in a measure, I succeeded, though I obviously learnt from him much more than he learnt from me."

Their most celebrated joint result was on the partition function, p(n), the number of ways a number n can be written as a sum of positive integers. The values explode as n grows; p(200) is a thirteen-digit number. Hardy and Ramanujan produced an asymptotic formula so sharp that five terms of it were enough to yield p(200) exactly.

The famous anecdote from this period belongs to the hospital, not the study. Ramanujan fell ill in 1917 and was never well again, but Hardy visited him through it. In Twelve Lectures Hardy recalled: "I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen." Ramanujan answered at once: "No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." (One cubed plus twelve cubed, and nine cubed plus ten cubed, both equal 1729.) Hardy added that it was Littlewood who said "every positive integer was one of Ramanujan's personal friends." The story has been polished by retelling, and scholars note that Ramanujan had already written down 1729's property in a notebook before he left India. The point is not discovery on the spot but the intimacy with numbers, the sense that to Ramanujan each integer had a personality.

The honors came despite the illness. He was elected to the London Mathematical Society in December 1917, to the Royal Society on 2 May 1918, proposed by Hardy and thirteen others, and to a six-year fellowship of Trinity College on 10 October 1918. Hardy later said, when the young Paul Erdős asked him his greatest contribution, that it was "the discovery of Ramanujan," and he kept an informal ability scale on which he scored himself a 25, Littlewood a 30, Hilbert an 80, and Ramanujan 100. He called the association "the one romantic incident in my life."

The last year, and the notebook that slept

Ramanujan's health had broken in the spring of 1917. He moved through nursing homes and sanatoria in Cambridge, Wells, Matlock, and London. The diagnosis was tuberculosis, with vitamin deficiency from a wartime vegetarian diet and the English cold. That diagnosis is now doubted. In 1994 a physician, D. A. B. Young, re-examined the records and concluded Ramanujan most likely died of hepatic amoebiasis, a parasitic infection of the liver arising from bouts of dysentery he had suffered in India in 1906 and 1909. It was endemic in Madras, could lie dormant for years, and would have been readily curable if recognized. By the time he sailed for home on 27 February 1919, he was a very sick man.

In India he did not recover. His wife Janaki Ammal, who had married him in 1909 as a child of ten and would outlive him by seventy-four years, cared for him. Late in life she told the mathematician Bruce Berndt what those last months were like. Ramanujan, she said, "continued 'to do his sums' until 4 days before he died when the pain became too great." He told her that going to England and working with Hardy "was the best thing that ever happened to him" and that, despite his failing health, "he had no regrets whatsoever." He was "extremely grateful to Hardy." On arriving home his first words to her had been: "If I had taken you with me, I would not have become ill."

His physician in Madras, Dr. P. S. Chandrasekhar, a tuberculosis specialist, kept a diary. The entry for 27 April 1920, the day after the death, is bitter. "Ramanujan's death, on April 26, 1920, could have been avoided. If he had been allowed to follow my instructions, this double tragedy need not have taken place." He recorded that Ramanujan "had lost the will to live on several occasions" and that "he shouldn't have returned to India." His death, the doctor wrote, "is the saddest paragraph" of the story.

Ramanujan was still producing mathematics. In his last letter to Hardy, dated 12 January 1920, he wrote: "I discovered very interesting functions recently which I call 'Mock' theta functions... they enter into mathematics as beautifully as the ordinary theta functions." It was a handful of examples, with the theory left unstated. He died on 26 April 1920, at the age of thirty-two. In his 1921 obituary notice, Hardy wrote that "his originality has been a constant source of suggestion to me ever since I knew him, and his death is one of the worst blows I have ever had." He also made the uneasy judgment that Ramanujan "would probably have been a greater mathematician if he had been caught and tamed a little in his youth," but that "he would have been less of a Ramanujan, and more of a European professor, and the loss might have been greater than the gain."

Then the papers went quiet. Janaki gathered the sheets of mathematics from his last year, with the slate he used, into the trunk he had brought from England. The pages passed from Janaki to Hardy, from Hardy to the mathematician G. N. Watson, who with B. M. Wilson began editing Ramanujan's notebooks and then, after Wilson died in 1935, let the project lapse. After Watson's own death in 1965, his papers, days from being burned, were rescued by J. M. Whittaker and R. A. Rankin, who sent Ramanujan's sheaf to the Wren Library at Trinity College on 26 December 1968. There it sat in a box, uncatalogued, for more than seven years.

In the spring of 1976 the American mathematician George Andrews, tipped off by Lucy Slater, visited the Wren Library to look through Watson's estate. The librarian's list included an item described as "A 139 page manuscript by S. Ramanujan on q-series." Andrews, who had written his doctoral thesis on the mock theta functions, knew at once what he was holding. The manuscript, more than a hundred sheets on 138 sides in Ramanujan's hand, held over six hundred formulas without proofs, and the full development of the mock theta functions he had only hinted at in his last letter. The discovery, its later editors wrote, has "frequently been deemed the mathematical equivalent of finding Beethoven's tenth symphony." Published by Narosa in 1987, the lost notebook is still being decoded, and its mock theta functions now reach into the entropy of black holes and string theory.

The romance of Ramanujan is not the romance of a life fully lived. It is the romance of a mind that produced, in perhaps six working years, more original mathematics than most gifted careers do in fifty, and kept producing, in pain and fever, four days from the end. Hardy's question, whether a fraud of genius was likelier than an unknown genius, was answered that night in 1913, and answered again every time a formula unread for half a century proved true.

Read the full life: The Man Who Knew Infinity: A Life of the Genius Ramanujan, by Robert Kanigel (Washington Square Press, 1991), the standard biography and the basis for the 2015 film, which places Ramanujan firmly in the Kumbakonam and Madras that made him.