My favorite paradox

kw: musings, logic

One of he famous stories of mathematics is about the "Hardy-Ramanujan Number" 1729. G.H. Hardy related:

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."

Of course, when Srinivasa Ramanujan said "number" of course he meant a natural number, not an integer, or he'd have said "integer". This is because, as his notebooks show, 91 is the sum of two different cubes two different ways, but one of the numbers being cubed is -5. Careful with his words, he was.

Suppose you wanted to make a list of interesting numbers (positive only, of course). You could start out with 1, the first number; 2, the first even number and the only even prime; 3, the smallest odd prime; 4, the smallest perfect square other than 1; 5, the number of digits on a primate hand, and the number that, when multiplied by any odd number, replicates itself as the last digit of the result; and so forth.

Sooner or later you might come to a number about which you know nothing "interesting", and you can't find anything about it. Have you found the first "uninteresting number"? How Interesting! Does it go on the list, or not?

By the way, these days you'd have to go far to find a number about which nothing has been written. Wikipedia includes thousands of pages about number after number. That first "unidentified flying number" probably has at least five digits.