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interesting numbers

Clearly some numbers are of greater interest (at least to mathematicians) than are others. The number pi, for instance, is far more interesting than 1.283 – or virtually any other number for that matter. Confining our attention to integers, can there be such a thing as an uninteresting number? It is easy to show that the answer must be "no." Suppose there were a set U of uninteresting integers. Then it must contain a least member, u. But the property of being the smallest uninteresting integer makes u interesting! As soon as u is removed from U, there is a new smallest uninteresting integer, which must then also be excluded. And so the argument could be continued until U was empty. Given that all integers are interesting can they be ranked from least to most interesting? Again, no. To be ranked as "least interesting" is an extremely interesting property, and thus leads to another logical contradiction!

When Srinivasa Ramanujan, the great Indian mathematician, was ill with tuberculosis in a London hospital, his colleague G. H. Hardy went to visit him. Hardy opened the conversation with: "I came here in taxi-cab number 1729. That number seems dull to me which I hope isn't a bad omen." Ramanujan replied, without hesitation: "Nonsense, the number isn't dull at all. It's quite interesting. It's the smallest number that can be expressed as the sum of two cubes in two different ways." (1729 = 13 + 123 and 93 + 103.)

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