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Sum of Two Cubes


Date: 01/12/2002 at 07:35:59
From: Gemm
Subject: Cube numbers

Find the smallest number that can be expressed as the sum of two cube 
numbers in two different ways


Date: 01/12/2002 at 17:05:48
From: Doctor Paul
Subject: Re: Cube numbers

The answer is 1729.

1729 = 1^3 + 12^3 = 9^3 + 10^3

What is more interesting about this is a celebrated story that goes 
with it. The great Indian mathematician Srinivasa Ramanujan 
(1887-1920) was ill in the hospital when a friend (the great British 
mathematician Godfrey Hardy) came to visit him. Hardy was well 
acquainted with Ramunujan's interest in numbers, so he noted that the 
number of the taxi he took to get the the hospital was rather 
uninteresting. The taxi number was, of course, 1729, and Ramanujan, 
though ill, immediately noted that 1729 was in fact quite interesting 
since it was the smallest number that could be written as a sum of two 
cubes in two different ways. 

How he had discovered this is anyone's guess. Computers have since 
verified that his answer was correct. Ramanujan became famous for 
coming up with formulas like this one off the top of his head.

I hope this helps.  Please write back if you'd like to talk about this 
more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School History/Biography
High School Number Theory

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