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Proof that there are no uninteresting numbers
By Graeme Mcrae on Tuesday, March 19, 2002 - 08:13 pm:

I wonder if it can be proven that there are no uninteresting numbers. Of course we must define "uninteresting" and "numbers" before we can hope to prove anything about them, so why don't we begin with the counting numbers, and say that such a number is "interesting" if it has a property that makes it unique. For example, 353 is "interesting" because it is the smallest multi-digit palindrome that is both prime and composed of entirely of prime digits. Any number about which a statement of this kind can't be made should be deemed "uninteresting".

Good; we've got that settled. Now for the proof:

Suppose there's an uninteresting number. Then, by the well-ordering principle, there is a smallest uninteresting number, which I'll call n. Being the smallest uninteresting number is itself quite interesting, don't you think? Since n alone has this disctinction, it is not uninteresting, a contradiction. Thus there are no uninteresting numbers.

By Niranjan Srinivas on Wednesday, March 20, 2002 - 10:50 am:

cool !!

By Sally Clough on Wednesday, March 20, 2002 - 01:47 pm:

My maths teacher showed me that one in 6th form. He was so fond of it he had a poster made :-)

By Niranjan Srinivas on Wednesday, March 20, 2002 - 04:45 pm:

is this not your "proof" than graeme ?
cheers
niranjan

By Yatir Halevi on Wednesday, March 20, 2002 - 09:05 pm:

Are you sure about this... In the same manner you can say that a number is interesting because it is the second non interesting number.
Lets say we have 2 uninteresting numbers than they are both interesting because they are the only ones. We can go on and add more numbers to the list, what I'm trying to say is where does the line draw? Lets say all the numbers are not interesting, but that fact makes them interesting because they are the only numbers which are uninteresting...

Do you see what I mean?

Yatir

By Brad Rodgers on Wednesday, March 20, 2002 - 09:34 pm:

Hmm...

I think the fact that every different number is in fact different would make for a pretty easy proof.

Brad

By Graeme Mcrae on Thursday, March 21, 2002 - 04:39 pm:

Niranjan, no, the idea is not original with me. I read about it in a book by Martin Gardner. The question of uninteresting numbers is similar to the "Paradox of the Unexpected Hanging" in that the important terms of the discussion are at best vague, and more likely completely meaningless.

In the Paradox of the Unexpected Hanging, in case you don't know that one, a prisoner is condemned to death, and his jailer, an amateur mathematician (as if there's any other kind), says "you will be hanged one day next week (Sunday through Saturday) and you won't be expecting it on that day."

The prisoner thought about this. Clearly the hanging couldn't take place on Saturday, because if the first six days elapsed without a hanging then he would expect it that day. So Saturday is out. That makes Friday the last possible day. But by the same reasoning, that day is out, too. Which makes Thursday the last day. The same reasoning rules out Wednesday, Tuesday, Monday, and, in fact, Sunday, too.

A smile came over the prisoner's face, because he realized the jailer could not keep his promise on any day, and would be forced to release him at the end of the week.

That smile faded on Wednesday morning when the hangman came to get him -- unexpectedly.

By Yatir Halevi on Friday, March 22, 2002 - 03:59 pm:

So what you are saying is that there is no two numbers with the same interesting property...

Yatir

By Kerwin Hui on Saturday, March 23, 2002 - 04:44 am:

There is a easier way to prove this given your definition of "interesting", since every natural number n has the unique property: it is expressible as the sum of n 1s.

Kerwin

By Yatir Halevi on Saturday, March 23, 2002 - 07:57 am:

Or it is in a distance d from an integer n.

Yatir

By Jane Millwood on Wednesday, July 24, 2002 - 10:27 pm:

Surely you wouldn't talk or write about a number if it were uninteresting? Since if you mention a number you are deemed to be talking or writing about it then it must therefore be interesting?