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I Happy monoid or semigroup?

  1. Jan 24, 2017 #1
    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
  2. jcsd
  3. Jan 24, 2017 #2


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    I don't get why ##13\circ 10 =1310## is happy. Shouldn't it be ##1^2+3^2+1^2+0^2=11 \rightarrow 1^2+1^2=2##? I wouldn't see a problem with the unity, as you could simply define ##\{\}## to be happy and ##a \circ \{\}=a\,.## It's no group because left-concatenation is no bijection (I guess).

    Edit: If you meant to continue: ##2 \rightarrow 2^2=4 \rightarrow 16 \rightarrow 37 \rightarrow 58 \rightarrow 89 \rightarrow 145 \rightarrow 42 \rightarrow 20 \rightarrow 4## which is a cycle without a ##1## in between.
  4. Jan 24, 2017 #3


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    I don't see an obvious closed operation between happy numbers, concatenation is not one.

    13, 23 and 1323 are all happy: there are examples where it works, but in general it does not.
  5. Jan 24, 2017 #4
    D'oh! Told you I didn't sleep. I had this funny feeling I would regret this post.

    Yeah, there might be something that works. But I should try again tomorrow.

    I'm going to hide under a rock now.
  6. Jan 24, 2017 #5


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    Don't be square. :cool:
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