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Using the definitions at Wikipedia; Sloane; and Mathworld; I can't see why $1$ is a member of the Kaprekar series?

Would someone give an easy explanation?


(Yet more on this here).

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Thanks DJC for the restoration of the links. I'm new so the spam filter (quite correctly) limited the number of links I could create). – Paddy3118 Jun 8 '11 at 5:51
Found what I was after: The Kaprekar numbers by D.E.Iannucci says 1 is included by fiat! – Paddy3118 Jun 8 '11 at 6:32
I don't see what an Italian auto manufacturer has to do with it. $1$ is included because it satisfies the definitions, as both answerers agree. – Gerry Myerson Jun 8 '11 at 6:47
up vote 2 down vote accepted

$1=0+1$, $1^2=0\times10^m+1$ seems to fit the definition as given at the OEIS reference.

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It also fits the definition given at Wikipedia. – JavaMan Jun 8 '11 at 0:32
If we set m to its minimum i.e. 1; q is 0, so r is 1. Then r is not <(10^1) ? – Paddy3118 Jun 8 '11 at 6:11
@Paddy3118, I don't understand. $10^1=10$, so, if $r=1$, then certainly $r\lt10^1$. – Gerry Myerson Jun 8 '11 at 6:44
Umm, your right. But I do like the Iannucci paper just stating it is 'cos it is :-) – Paddy3118 Jun 8 '11 at 6:55

Unfortunately, my answer is anticlimactic. I happen to know wikipedia's definition (linked in the question, but I reproduce the definition here)

Let X be a non-negative integer. X is a Kaprekar number for base b if there exist non-negative integers n, A, and positive number B satisfying: $X^2 = Ab^n + B$, where $0 < B < b^n$, and s.t. $X = A + B$

So $A$ can be $0$. Thus $1^2 = 1 = 0* 10^1 + 1$, and we see that it's a Kaprekar number.

And - Gerry posted his answer just before me (I refreshed, and it's there)! But I wrote this too, so I'll keep it -

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There is no need to sign your name at the end, since it appears at the bottom anyway. – Eric Naslund Jun 8 '11 at 0:42
Unfortunately the WP article goes on to state the 0<B<N with the minimum N being 1 so B cannot be 1 ? (the range on OEIS is slightly different - and in the WP nomenclature becomes 0<=B<N). – Paddy3118 Jun 8 '11 at 6:02
@Paddy - this doesn't really matter, as it could just be 2 instead. Then we still see that 1 is such an example, as A is 0. That's the important part. – mixedmath Jun 8 '11 at 6:12
Yep. I now see where I was wrong. Thanks. – Paddy3118 Jun 8 '11 at 6:57

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