[1eyedjack]

Not a book for those with significant formal training in mathematics - y'all will know it all already. But for me, I found Alex Bellos's book Alex's Adventures in Numberland an amusing read (Bloomsbury, ISBN 978-0-7475-9716-2).

I also knew quite a bit of the subject matter, but it didn't stop it being an enjoyable read, and it brought home to me how much more interesting "pairs of numbers" might be than simply "interesting numbers".

For example, who would have thought that
2592
and
24547284284866560000000000
would make an "interesting pair"?

[EDIT] Actually, having Googled the larger number, I find that it is really pretty boring. There are another 10 numbers in the set, being the integers in the range 0 to 9 inclusive.

[gwnn]

24547284284866560000000000 wow.

[kenberg]

I may get this book! Is that sick or what?

I have told my wife she may not see me for a while, I am off on a quest for indestructible numbers in base 12.

[Gerben42]

As it can be proven that all integers are interesting, I would assume the book is rather heavy...

(just kidding, I read the book when I was a student and didn't break my back in the process)


P.S. The proof: Let 0,1,2,... be denoted as the natural numbers. Zero has the property that it is the smallest natural number, so it is interesting. Then suppose n > 0 is the smallest uninteresting number. As this would be an interesting property, we have a contradiction. Ergo all natural numbers are interesting.

Extension to integers: Consider the same logic applied to the sequence 0, 1, -1, 2, -2, 3, -3, ...

Home exercise: Prove that all rational numbers are interesting.

[wyman]

Incidentally, there are other indestructible numbers (e.g., 9225 or 92251). The author of this book has things incorrect, slightly.

The actual conjecture of Sloane is that 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2592, 24547284284866560000000000 are the only fixed points of the powertrain map (A point x is fixed by a map F if F(x)=x).

I have not seen it in print, but it may also be a conjecture that every integer other than those above has a fixed point in its orbit:
that is: in the set {x, F(x), F(F(x)), F(F(F(x))), ...} one of the above numbers always comes up.

Note that it's not immediately obvious that there aren't distinct x & y, such that F(x) = y and F(y) = x.
Nor is it obvious (to me) that there aren't distinct x, y, z such that F(x) = y, F(y) = z, and F(z) = x.
[Maybe it is clear, but I haven't thought about it]