Definition:Kaprekar Number

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Definition

Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Suppose that $n^2$, when expressed in number base $b$, can be split into two parts that add up to $n$.

Then $n$ is a Kaprekar number for base $b$.


Sequence of Kaprekar Numbers

The sequence of Kaprekar numbers begins:

$1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, \ldots$


Some sources do not include such numbers as $4879$ and $5292$:

$1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, \ldots$

where the $2$nd of the $2$ parts begins with one or more leading zeroes:

$4879^2 = 23 \, 804 \, 641 \to 238 + 04641 = 4879$
$5292^2 = 28 \, 005 \, 264 \to 28 + 005264 = 5292$


Examples of Kaprekar Numbers

$142 \, 857$ is Kaprekar

\(\displaystyle 142 \, 857^2\) \(=\) \(\displaystyle 20 \, 408 \, 122 \, 449\)
\(\displaystyle 20 \, 408 + 122 \, 449\) \(=\) \(\displaystyle 142 \, 857\)


$1 \, 111 \, 111 \, 111$ is Kaprekar

\(\displaystyle 1 \, 111 \, 111 \, 111^2\) \(=\) \(\displaystyle 1 \, 234 \, 567 \, 900 \, 987 \, 654 \, 321\)
\(\displaystyle 123 \, 456 \, 790 + 0 \, 987 \, 654 \, 321\) \(=\) \(\displaystyle 1 \, 111 \, 111 \, 111\)


$22 \, 222 \, 222 \, 222 \, 222$ is Kaprekar

\(\displaystyle 22 \, 222 \, 222 \, 222 \, 222^2\) \(=\) \(\displaystyle 493 \, 827 \, 160 \, 493 \, 817 \, 283 \, 950 \, 617 \, 284\)
\(\displaystyle 4 \, 938 \, 271 \, 604 \, 938 + 17 \, 283 \, 950 \, 617 \, 284\) \(=\) \(\displaystyle 22 \, 222 \, 222 \, 222 \, 222\)


Also see


Source of Name

This entry was named for Dattathreya Ramchandra Kaprekar.


Sources