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© Copyright 2000, Jim Loy
The following is a famous story told by G.H. Hardy about S. Ramanujan, repeated word for word in various sources:
Once, in a taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, "rather a dull number," adding that he hoped that wasn't a bad omen. "No, Hardy," said Ramanujan, "it is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways."
To be accurate, Ramanujan should have said "the sum of two positive cubes in two different ways." 1729 has since become known as the Hardy-Ramanujan Number, even though this feature of 1729 was known more than 300 years before Ramanujan. And numbers of its type (the smallest numbers expressible as the sum of 2 cubes in n ways) are sometimes called Taxicab Numbers. 1729=13+123=93+103.
How did Ramanujan know this? Well, he loved numbers. All through his life, he experimented with numbers. As a child, he probably found prime numbers into the many thousands, trying to find a pattern that other people had missed. 1729 is not a prime, by the way. Such a person also examines Pythagorean triples, like 3,4,5 which are of the form 32+42=52. He would search for patterns in these numbers, too. Perhaps he found an equation to generate all possible Pythagorean triples, as such equations exist. Not all numbers can be expressed as the sum of two squares. He probably found numbers that were expressible as the sum of two squares in two or more different ways. It is natural that he examined the sums of cubes (and higher powers) in the same way. He would notice that 1729 came up twice on this list, and he would remember it all his life.
By the way, equations such as c=a3+b3 are called Diophantine Equations, named after Diophantus (sometime in the period 200 BC - 400 AD, probably about 250 AD), when they involve only integers.