The Man Who Knew Infinity

This article is all about Srinivasa Ramanujan who had almost no formal training in pure mathematics, still got FRS for his contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems those considered unsolvable at that time. 

During his short life (1887-1920), Ramanujan independently compiled nearly 3,900 results (mostly identities and equations). Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae, and mock theta functions, have opened entire new areas of work and inspired a vast amount of further research. The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan, and his notebooks containing summaries of his published and unpublished results have been analyzed and studied for decades since his death as a source of new mathematical ideas. As late as 2012, researchers continued to discover that mere comments in his writings about "simple properties" and "similar outputs" for certain findings were themselves profound and subtle number theory results that remained unsuspected until nearly a century after his death. He became one of the youngest Fellows of the Royal Society and only the second Indian member, and the first Indian to be elected a Fellow of Trinity College, Cambridge. His lost notebook, containing discoveries from the last year of his life, caused great excitement among mathematicians when it was rediscovered in 1976.

He once said, "An equation for me has no meaning unless it expresses a thought of God."

The Hardy–Ramanujan number 1729

The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy's words: "I remember once going to see him when he was ill at Putney. 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 unfavorable 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."

The two different ways are:
1729 = 1³ + 12³ = 9³ + 10³

The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729):

91 = 6³ + (-5)³ = 4³ + 3³

Generalizations of this idea have created the notion of "taxicab numbers".

Immediately before this anecdote, Hardy quoted Littlewood as saying, "Every positive integer was one of [Ramanujan's] personal friends."