S ramanujan

Ramanujan's manuscript. The representations of 1729 as the sum of two cubes appear in the bottom right corner. The equation expressing the near counter examples to Fermat's last theorem appears further up: α 3 + β 3 = γ 3 + (-1) n. Image courtesy A box of manuscripts and three notebooks. That's all that's left of the work of Srinivasa Ramanujan, an Indian mathematician who lived his remarkable but short life around the beginning of the twentieth century. Yet, that small stash of mathematical legacy still yields surprises. Two mathematicians of Emory University, Ramanujan's story is as inspiring as it is tragic. Born in 1887 in a small village around 400 km from Madras (now Chennai), Ramanujan developed a passion for mathematics at a young age, but had to pursue it mostly alone and in poverty. Until, in 1913, he decided to write a letter to the famous Cambridge number theorist The taxi-cab number The romanticism rubbed off on the number 1729, which plays a central role in the Hardy-Ramanujan story. "I remember once going to see [Ramanujan] when he was ill at Putney," Hardy wrote later. "I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. 'No', he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'" What Ramanujan meant is that The anecdote gained the number 1729 fame in mathematical circles, but until re...

SRINIVASA RAMANUJAN Srinivasa Ramanujan It is one of the most romantic stories in the history of mathematics: in 1913, the English mathematician G. H. Hardy received a strange letter from an unknown clerk in Madras, India. The ten-page letter contained about 120 statements of theorems on infinite series, improper integrals, continued fractions, and number theory (Here is a Thus was Srinivasa Ramanujan (1887-1920) introduced to the mathematical world. Born in South India, Ramanujan was a promising student, winning academic prizes in high school. But at age 16 his life took a decisive turn after he obtained a book titled A Synopsis of Elementary Results in Pure and Applied Mathematics. The book was simply a compilation of thousands of mathematical results, most set down with little or no indication of proof. It was in no sense a mathematical classic; rather, it was written as an aid to coaching English mathematics students facing the notoriously difficult Tripos examination, which involved a great deal of wholesale memorization. But in Ramanujan it inspired a burst of feverish mathematical activity, as he worked through the book's results and beyond. Unfortunately, his total immersion in mathematics was disastrous for Ramanujan's academic career: ignoring all his other subjects, he repeatedly failed his college exams. As a college dropout from a poor family, Ramanujan's position was precarious. He lived off the charity of friends, filling notebooks with mathematical discover...

The unlikely tale of a mysterious letter, and its place in the history of mathematics This week’s release of the movie The Man Who Knew Infinity (which I saw in rough form last fall through its mathematican-producers Manjul Bhargava and Ken Ono ) leads me to write about its subject, Srinivasa Ramanujan … “Hardy” and “Ramanujan” in The Man Who Knew Infinity #### A Remarkable Letter They used to come by physical mail. Now it’s usually email. From around the world, I have for many years received a steady trickle of messages that make bold claims — about prime numbers, relativity theory, AI, consciousness or a host of other things — but give little or no backup for what they say. I’m always so busy with my own ideas and projects that I invariably put off looking at these messages. But in the end I try to at least skim them — in large part because I remember the story of Ramanujan. On about January 31, 1913 a mathematician named Empirical math in Ramanujan’s letter (Reproduced by kind permission of the Syndics of Cambridge University Library)But some things get more exotic, with pages of formulas like this: Remarkable formulas in Ramanujan’s letter (Reproduced by kind permission of the Syndics of Cambridge University Library)What are these? Where do they come from? Are they even correct? The concepts are familiar from college-level calculus. But these are not just complicated college-level calculus exercises. Instead, when one looks closely, each one has something more exotic a...

Throughout the history of mathematics, there has been no one remotely like Srinivasa Ramanujan. There is no doubt that he was a great mathematician, but had he had simply a good university education and been taught by a good professor in his field, we wouldn’t have As the years pass, I admire more and more the astonishing body of work Ramanujan produced in India before he made contact with any top mathematicians. Not because the results he got at the time changed the face of mathematics, far from it, but because, working by himself, he fearlessly attacked many important and some not so important problems in analysis and, especially, number theory – simply for the love of mathematics. It cannot be understated, however, the role played by Ramanujan’s tutor Godfrey Harold Hardy in his life story. The Cambridge mathematician worked tirelessly with the Indian genius, to tame his creativity within the then current understanding of the field. It was only with Hardy’s care and mentoring that Ramanujan became the scholar we know him as today. Srinivasa Ramanujan. Determined and obsessed In December 1903, at the age of 16, Ramanujan passed the matriculation exam for the University of Madras. But as he concentrated on mathematics to the exclusion of all other subjects, he did not progress beyond the second year. In 1909 he married a nine-year-old girl, but failed to secure any steady income until the beginning of 1912, when he became a clerk in the Madras Port Trust office on a meagr...