## Cover Story

### Raiders of the Lost Notebook

LAS mathematician tracks proof for legendary numbers genius.

The "Lost Notebook" sounds like something that Indiana Jones would pursue through jungles and across sands. But for mathematicians worldwide, the quest for this rare gem has taken them on a much different path—through libraries and stacks and stacks of documents.

Bruce Berndt, a College of Liberal Arts and Sciences mathematician, never had to outrun boulders or dangle above a pit of poisonous snakes as did Indiana Jones. Nevertheless, he has pursued the secrets of the "Lost Notebook" for more than 10 years, digging through this mathematical treasure with the determination of any archeological adventurer.

In 2005, Berndt and G.E. Andrews, the man who uncovered the "Lost Notebook", published the first of several volumes of books, in which they provide proofs for the formulas found within the famous notebook.

The "Lost Notebook" contains more than 600 mathematical formulas written by one of the greatest mathematical minds in history—Srinivasa Ramanujan, an Indian scholar who grew up in poverty, received little formal training, and died at the age of 32.

#### Revered Like Einstein

Although Ramanujan is unknown throughout the West, except in mathematical circles, "the recognition of Ramanujan's name in India can perhaps only be compared with the recognition that the names of Newton and Einstein have in Western culture," Berndt says. "In India today, Ramanujan is revered more than any other Indian scientific figure."

Berndt's co-writer, Andrews, discovered the "Lost Notebook" in 1976 while sifting through papers left to the Trinity College Library at Cambridge University. As Berndt puts it, "The discovery of this ‘Lost Notebook' caused roughly as much stir in the mathematical world as the discovery of Beethoven's tenth symphony would cause in the musical world."

So who was this mathematician, who inspired two movies about his life from India, a third from England, and a fourth now in the planning stages in Hollywood?

Ramanujan was born in 1887 and grew up in a one-room home in Kumbakonam, India. As a 12-year-old prodigy, he displayed uncommon abilities when he borrowed S.L. Loney's *Plane Trigonometry* and proceeded to work out every problem in the book—with no help.

At the age of 16, Ramanujan received a scholarship to attend Government College at Kumbakonam. But he was so single-mindedly devoted to mathematics that he would not study any other subject. As a result, he failed most of his classes, except for mathematics, and lost his scholarship. The same thing happened at the second school he attended.

Committed to the social and religious traditions of India, the 22-year-old Ramanujan then married a nine-year-old bride picked out for him by his parents. But the new marriage, which took place in 1909, meant he needed an income. So he found a job as a clerk at the Madras Port Trust Office, which was a turning point in his life, according to Berndt.

Ramanujan put in grueling hours, working all day at the office and then going home to do math through the night—until 6 a.m. the following day. After a few hours of sleep, he was back at the office. The job was tough, but the office manager, who was also a mathematician, took a keen interest in his "side" research.

Upon the urging of his employer, Ramanujan sent his work to several noted mathematicians. Many of them wrote him off as a crank, but one of the world's leading mathematicians, G.H. Hardy, did not. After initial skepticism about this mysterious mathematician, Hardy was astounded by Ramanujan's theorems. As he wrote, "I had never seen anything in the least like them before…. They must be true because, if they were not true, no one would have had the imagination to invent them."

#### Collaborating at Cambridge

Hardy invited Ramanujan to Cambridge. But being part of a conservative branch of Brahminism, Ramanujan resisted the idea of traveling to England because overseas travel would make him "unclean." According to stories, Ramanujan's mother had a dream in which she saw her son surrounded by influential Europeans, and she took this as a sign that her son should be allowed to travel to Cambridge.

So Ramanujan struck off for Cambridge in 1914; but like many visiting scholars from India, he left behind his young wife. (The couple didn't have children and never would.) For the next five years, Ramanujan and Hardy struck up a historical, five-year collaboration.

"Hardy and Ramanujan discussed mathematics daily and jointly wrote some monumental papers in the theory of numbers," Berndt says. "During Ramanujan's first three years at Cambridge, he gained worldwide fame for his mathematical findings."

Unfortunately, within three years of reaching Cambridge, Ramanujan also contracted the illness that would later kill him.

Mathematician Bruce Berndt has devoted nearly three decades to providing proofs to the 3,904 formulas contained in the botebooks of Srinivasa Ramanujan, one of India's-and the world's-greatest mathematicians.

Berndt says Ramanujan's illness had about five different diagnoses, ranging from tuberculosis to a poor diet. However, about 10 years ago, an English physician examined all of the records of Ramanujan's illness and concluded that the young mathematician died of hepatic amoebiasis, an amoebic infection of the liver that can result if dysentery is not treated properly.

"It appears that Ramanujan had two bouts of dysentery before he left for England," Berndt says. "If dysentery is not treated properly, the amoebae can live in the large intestinal area for years and not affect the patient. But if one's lifestyle changes, as Ramanujan's clearly did when he arrived in England, the amoebae can become active and eventually kill the patient."

In the midst of the ill health and heavy workload, Ramanujan maintained a good sense of humor. According to Berndt, Ramanujan was talkative and loved to tell jokes. But most people never heard him tell a joke from start to finish because he usually started laughing uncontrollably before he got to the end.

Ramanujan returned to India in 1919, but he never recovered from his illness. When he died on April 26, 1920, he remained immersed in the world of numbers until the very end. In fact, Ramanujan's widow told Berndt that her husband continued to work on math problems until four days before he died.

#### How the Notebook Was Lost

Since the age of 16, Ramanujan had been steadily filling up notebooks with mathematical formulas. But the "Lost Notebook" refers specifically to the notebook that he wrote in the final year of his life.

Despite its name, the "Lost Notebook" actually is not a notebook at all. It is a hodgepodge of papers, on which Ramanujan wrote mathematical formulas. The "Lost Notebook," Berndt says, "represents possibly his deepest work. Thus, even though he worked in great pain, with his physical strength ebbing away, Ramanujan's creativity did not diminish but rather gained in strength."

After his death, Ramanujan's materials followed a circuitous path, which is when the "Lost Notebook" became lost. The papers were first shipped to the University of Madras library, which later sent many of the materials to Hardy for publication. Then sometime between the late 1930s and late 1940s, Hardy passed on the "Lost Notebook" to the English mathematician G.N. Watson.

The "Lost Notebook" sat in Watson's office for years until he died in 1965. That's when a fellow mathematician found the "Lost Notebook" among Watson's papers, which covered the floor of his large office to the depth of a foot. This mathematician saved the papers just days before they were scheduled to be burned. However, not realizing the significance of Ramanujan's papers, he simply sent them to Trinity College at Cambridge.

The "Lost Notebook" finally surfaced eight years later, when Andrews stumbled across it in 1976.

#### Proving Ramanujan's Formulas

Berndt's own fascination with Ramanujan began in 1974 when he discovered that by using one of his own theorems he could prove some of the formulas in the notebooks that Ramanujan had written before the "Lost Notebook." Ramanujan had never written down his proofs, partly because of the shortage of paper, so the task of proving the formulas had loomed as an unfinished task in the mathematical world. (Mathematicians use proofs as a critical step in checking the accuracy of their reasoning.)

Berndt wondered whether there were other formulas of Ramanujan's that he could prove. So, what began with simple curiosity ultimately led to 20 years of steady work, whittling away at the formulas. With the help of many other mathematicians, Berndt managed to solve all of the 3,254 formulas in Ramanujan's early notebooks. Out of those, he found that only a few formulas—"less than five"—were wrong.

After Berndt published these results in five books, he and Andrews began tackling the 600-plus formulas in the "Lost Notebook," some of which were scrawled haphazardly across papers. To get an idea of what's involved in proving these formulas, it took one of Berndt's graduate students 180 pages to prove just four formulas from the "Lost Notebook." Berndt thinks Ramanujan probably could have done proofs for the same four formulas in just a few pages. But no one knows for sure since Ramanujan didn't write down his proofs.

Ramanujan was known as a number theorist, Berndt says, but his work has influenced many other areas. Ramanujan was clearly a mathematical diamond in the rough with an undying passion for numbers and their properties. For instance, Hardy once recalled the time he visited Ramanujan in a nursing home, when the mathematician was in his final days.

"I rode here today in taxicab number 1729," Hardy told Ramanujan. "This seems to be a dull number and I hope it's not an unfavorable omen." To which Ramanujan replied, "No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways."

For Ramanujan, no number was dull. Numbers were always full of surprises.

As Berndt puts it, "Generally, when mathematicians see a published paper, they might marvel at the math, but the results are not really surprising. But for Ramanujan, a lot of his mathematics is surprising. One gets the feeling that if he hadn't discovered a particular result, then no one would have discovered it—not even 100 to 200 years later."

Read more in "To Infinity and Beyond."

By Doug Peterson

Fall/Winter 2006-07