Srinivasa Ramanujan · Mathematics
1729 as the smallest number that is a sum of two cubes in two different ways - the famous hospital story - together with Ramanujan’s deep work on highly composite numbers and the theory of the partition function, all expressions of his intimacy with the whole numbers.
The most famous anecdote in modern mathematics is small and true. During Ramanujan’s years in England he fell seriously ill, and Hardy went to visit him in a nursing home at Putney. Casting about for something to say, Hardy remarked that he had arrived in taxicab number 1729, which struck him as a rather dull number, and he hoped it was not a bad omen. Ramanujan answered at once that 1729 was not dull at all: it was the smallest number expressible as the sum of two cubes in two different ways. And so it is: 1729 equals 1 cubed plus 12 cubed, and also equals 9 cubed plus 10 cubed, and no smaller number can be split into two cubes in two distinct ways. Where Hardy saw a forgettable integer, Ramanujan saw, instantly and without calculation, a hidden double structure. The story is beloved because it captures in miniature the thing that set Ramanujan apart: for him the whole numbers were not a grey uniform crowd but a gallery of individuals, each with its own secret face.
That same intimacy drove Ramanujan’s serious research into the fine structure of the integers. One of his major early works, a long paper of 1915, studies what he called highly composite numbers: numbers that have more divisors than any smaller number. The number 12 is highly composite, since it has six divisors (1, 2, 3, 4, 6, 12) and no number below it has as many; so are 24, 36, 48, 60, and a sparse, ever-growing list beyond. Ramanujan analysed exactly how such record-breaking numbers are built from prime factors, and how the number of divisors can be made as large as possible for a given size. The paper was so long and detailed that the London Mathematical Society, short of funds, asked him to cut it, and a large part went unpublished for decades. It remains a classic, the founding study of a whole topic, and a monument to his feel for the multiplicative anatomy of numbers.
Ramanujan’s love of the integers reached its height in his work on the partition function p(n), the count of ways to write n as a sum of positive whole numbers. Beyond the great asymptotic formula he found with Hardy, Ramanujan discovered something even more mysterious hiding in the partition numbers: exact patterns of divisibility, now called the Ramanujan congruences. He noticed, and proved, that the number of partitions of any number ending in 4 or 9 is always divisible by 5; that p(n) is divisible by 7 whenever n leaves remainder 5 on division by 7; and a similar rule for 11. There is no obvious reason the chaotic-seeming partition counts should obey such clean arithmetic laws, yet they do, and Ramanujan saw it. These congruences opened a deep field connecting partitions to modular forms, and mathematicians are still finding new ones. Once again the pattern holds: Ramanujan stared at the integers until they confessed a secret regularity no one else had noticed.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that 1729 is the smallest number expressible as a sum of two cubes in two ways, that Ramanujan saw this instantly from his hospital bed, and that he did deep work on highly composite numbers and on partitions. Explain what these have in common about how Ramanujan ‘knew’ numbers.
Leads to Diophantus of Alexandria.
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