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My First Theorem

It's not everyday that I get to exercise my brain in a theoretical fashion, but today was special. I suppose a cosmic ray zapped that one neuron that was holding the rest of them back. Or maybe that cosmic ray zapped too many of those neurons, and now I am delusional about my own grandeur.

Anyway, I have a theory about numbers. Take any integer number and add up its digits. That means for a number like '123' you add '1 + 2 + 3'. My first theory is that the resulting sum will have no more than N-1 digits, where N is the number of digits in the original number. That's nothing new. Call this the DSUM operator.

Now for my conjecture:
(1) Given any number, X, define Y = DSUM(X).
(2) For every Y[i] = DSUM(Y[i-1]), the number of digits in Y[i] will be less than in Y[i-1].
(3) [i] will always be less than or equal to N, where N is the number of digits in X.
(4) For all Y[i], Y[i] is always less than Y[i-1].
(5) There always exists a value of Y[i] such that Y[i] is less than 10.

Now let's do some examples.

X = 123
Y[1] = 1 + 2 + 3 = 6

X = 94567
Y[1] = 9 + 4 + 5 + 6 + 7 = 31
Y[2] = 3 + 1 = 4

X = 0.13598
Y[1] = 1 + 3 + 5 + 9 + 8 = 26
Y[2] = 8

For fractional numbers, (4) does not hold, so it has to be inverted. Y[i] is always greater than Y[i-1].


Yeah fun with numbers. So you read it here first, and I'm off to be famous.

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