Link o'the week

Every number has an interesting property

I have often heard the story about G. H. Hardy’s meeting with S. Rumanujan at the hospital.   According to Hardy:

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.”

Related to this is  the following joke (which I have been told by students looking to get  a bit of extra credit by showing their mastery of basic proof techniques):

Theorem: Every natural number has  an interesting property.

Proof. Assume the contrary. Let A be the least natural number with no interesting property whatsoever.  Well, that’s a pretty interesting property about A, so we have a contradiction. Q.E.D.

Logical errors aside, I often find myself looking at natural numbers and trying to find their “interesting” property.

Well, look no more.   Erich Friedman has conveniently listed  interesting  properties for the first ten thousand natural numbers, starting at 0.

Astute minds will note that ten thousand examples does not a proof make.   Well, fret not, ye of sharp mental acumen.   Mike Keith (author of the astounding  Cadaeic Cadenza) gives a neat, direct proof that every number has an interesting property by proving that every natural number is the only number to satisfy being expressed as a sum of a certain number of   powers in a certain number of ways.   Enjoy the cleverness!’

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