Hardy-Ramanujan number

One thousand, seven hundred and twenty nine.

The story goes that the English professor of mathematics G. H. Hardy was once traveling to see his student Srinivasa Ramanujan. As Hardy relates:

I remember once going to see him while he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to me rather a dull one, and I hoped it was not an unfovourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways."

The story is used as an example of the remarkable mental powers possessed by Ramanujan.

This question cropped up once on University Challenge: "What is the smallest number expressible as a sum of two cubes in two different ways?" Naturally, some smartypants immediately buzzed and answered 1729, treating the delighted viewer to probably the first time in history that Jeremy Paxman had been rendered speechless. As I recall his eventual retort was along the lines of, "bloo... dy... hell".

I remember once going to see him [Ramanujan] when he was lying 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."

G. H. Hardy in Ramanujan (London 1940)
http://www-gap.dcs.st-and.ac.uk/~history/Quotations/Hardy.html

This event took place in 1917 when Hardy went to visit Ramanujan in the hospital. Srinivasa Ramanujan was initially a clerk in India when he wrote to Hardy who was one of the most famous mathematicians in the world (if not the most famous - if only until Ramanujan gained some recognition) asking Hardy to examine some of his work. Hardy was most impressed and arranged for the young Ramanujan to attend Cambridge in 1913. In 1917, Ramanujan fell ill and was in the hospital much of the time. Being a strict vegetarian he attributed his health problems to the English food, its poor quality, and food shortages with the outbreak of World War I and so returned to India in 1919. Unfortunately his health did not improve and Ramanujan died of tuberculosis in April 26, 1920.

It turns out that 1729 is not a dull number at all (to a mathematician). First It has the identity that is mentioned - being the smallest counting number that was represented as sum of a two cubes in two different ways:

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

Integer solutions to:

 3    3    3    3
I  + J  = K  + L

are now known as Ramanujan numbers. Other such Ramanujan numbers include 4104 (2,16,9,15), 13832 (18,20,2,24), 20683 (10,27,19,24), and 32832 (18,30,4,32). There are an infinite number of Ramanujan numbers.

1729 is also a Carmichael Number - a pseudoprime relative to every base. These are odd composite numbers that satisfy Fermat's little theorem.

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http://home.att.net/~s-prasad/math.htm
http://www.mathpages.com/home/kmath028.htm
http://www.durangobill.com/Ramanujan.html
http://mathworld.wolfram.com/CarmichaelNumber.html