What are 1728 and 1729?

Question

This is a rather open-ended question. The number 1729 number seems to come up more than a four digit number should, so I thought it might be fun to collect a few interesting properties of it. The most famous is probably that 1729 is the smallest number that can be written as the sum of two positive cubes in two different ways. This question was inspired by a seminar talk where "1728" kept showing up (elliptic curves).

Ian H · Accepted Answer

1728 is 12^3

1729 is known as the taxicab number.
British mathematician G. H. Hardy visited the Indian mathematician Srinivasa Ramanujan in hospital. 
He rode in taxi cab number 1729 and remarked that the number seemed to me rather dull 
Ramanujan 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 these:
1729 = 1^3 + 12^3 = 9^3 + 10^3

See WIKI for more taxicab type numbers expressible as the sum of two cubes in two different ways
1729  is also 7*13*19

Regards - Ian

Jacinta · Answer

Entirely irrelevant (but I'm waiting for the jug to boil): 

17,281,729 is a prime number

So is 28,172,917

Ditto 1,723,172,517,271,729 
(Grouped in fours: 1723 1725 1727 1729)

And 1,723,172,417,251,726,172,717,281,729
(Grouped in fours: 1723 1724 1725 1726 1727 1728 1729)

Coffee time :)

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What are 1728 and 1729?

2 Answers