Wow ... 68,
another year passing by and
it sounds like it is gradually getting later.
Apart from the obvious that it is a natural and an even number following 67 and preceding 69 Wikipedia tells me that 68 is a Perrin number.
It also happens to be the largest known number to be the sum of two primes in exactly two different ways, though it seems we can't be quitte so sure of that because of the Goldbach conjecture.
Furthermore it seems to be the case that a 68-sided regular polygon may be constructed with compass and straightedge, that is because of the factorization as you probably would have guessed.
If that doesn't fill your cup you might want to dive into 10-bit binary numbers of which there are exactly 68 in which each bit has an adjacent bit with the same value.
But then we would have to talk about combinatorially distinct triangulations and intervals in the Tamari lattice to describe the ways of parenthesizing five items. As I am not a mathimatician, I had never before heard of Størmer's theorem and I haven't the slightest idea what I wrote down here just now, I'd rather not talk about it and I had better not argue with any of the above.
Still it is true what they say: time flies!
Maybe I just might want to add/mention/quote that the largest graceful graph on 13 nodes has exactly 68 edges. But that is because I think 13 is also a very nice number, Wouldn't you agree and that today I completed exactly 68 years on this planet Earth.
I am a fan of Wikipedia though on some scores it may be a bit too informative for me. Anyway I was glad to read that 68 is a so-called Happy number and hope this next year will turn out to be a happy one. Cheers!