- #1
dkotschessaa
- 1,060
- 783
I haven't slept in awhile and I might have just come up with a totally useless or vacuous concept. It could possibly be a cool example of something. For some reason, I really like happy numbers.
A happy number is a number such that when you separate the digits, square each, and add them back together, you get the number 1 in a finite number of steps. i.e.
13 --> 1^2 + 3^2 = 1 + 9 = 10
10 --> 1^2 + 0+2 = 1
It follows then that if you concatenate two happy numbers you'd get another happy number. So this set is closed under concatenation. Let ## * ## be concatenation.Example:
13*10 = 1310 (which is clearly happy).
It's associative, and commutative. I suppose since I am using concatenation that the identity element is just the empty word. But the empty word isn't a number. It's certainly not a group since there is no inverse.
Without the identity it's at least a commutative semigroup. Kind of interesting. Or maybe not.
-Dave K
A happy number is a number such that when you separate the digits, square each, and add them back together, you get the number 1 in a finite number of steps. i.e.
13 --> 1^2 + 3^2 = 1 + 9 = 10
10 --> 1^2 + 0+2 = 1
It follows then that if you concatenate two happy numbers you'd get another happy number. So this set is closed under concatenation. Let ## * ## be concatenation.Example:
13*10 = 1310 (which is clearly happy).
It's associative, and commutative. I suppose since I am using concatenation that the identity element is just the empty word. But the empty word isn't a number. It's certainly not a group since there is no inverse.
Without the identity it's at least a commutative semigroup. Kind of interesting. Or maybe not.
-Dave K