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1729 = 2 Cubes 2 Ways

Ramanujan’s 129th Birthday

By Monty, 22nd December 2016

Someone on Reddit pointed out that today (Dec 22nd) is Ramanujan’s 129th birthday. One of the commenters further pointed out that 129 is the sum of the first 10 primes, and the smallest number that is the sum of three squares in four different ways (echoing Ramanujan’s comment on G.H. Hardy’s taxi number 1729 being the smallest number which is expressible as the sum of two cubes in two different ways). The first claim is easily checked by adding up the first 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. The second claim is less obvious so I decided to write a Python program to check it:

Hardy-Ramanujan Numbers

1729 is the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a visit to the hospital to see the Indian mathematician Srinivasa Ramanujan. 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

What do you think?

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