Some numbers are more famous than others
Find the smallest integer that can be written as the sum of two cubes in exactly two different ways. Why is it famous?
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June 25th, 2009 at 1:41 am
91 = 3³ + 4³ = (-5)³ + 6³.
It’s famous as a counterexample to this puzzle.
June 25th, 2009 at 11:39 am
Ah, well, in that case,
3 = (cube root of 3)³ + 0³ = (cube root of 2)³ + 1³.
June 25th, 2009 at 11:53 am
What is the answer to the puzzle if ‘cubes’ have to be taken from the set 1, 8, 27, 64, …?
June 25th, 2009 at 9:29 pm
1³+12³=9³+10³
June 25th, 2009 at 9:37 pm
In trying to answer why this is a famous number, I googled “1729″ and found this page: http://en.wikipedia.org/wiki/1729_(number). As far as I can tell, 1729 seems to only be famous for the property noted in this forum.
June 25th, 2009 at 9:38 pm
The link is messed up. Sorry — you’ll need to copy / paste the full url for wikipedia to come up with anything.
June 25th, 2009 at 10:07 pm
1729 is known as the Hardy–Ramanujan number after a famous anecdote of the British mathematician G. H. Hardy regarding a hospital visit to 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.”
Taken from the Wikipedia link above.