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
\(\ds 142 \, 857^2\) | \(=\) | \(\ds 20 \, 408 \, 122 \, 449\) | ||||||||||||
\(\ds 20 \, 408 + 122 \, 449\) | \(=\) | \(\ds 142 \, 857\) |
$1 \, 111 \, 111 \, 111$ is Kaprekar
\(\ds 1 \, 111 \, 111 \, 111^2\) | \(=\) | \(\ds 1 \, 234 \, 567 \, 900 \, 987 \, 654 \, 321\) | ||||||||||||
\(\ds 123 \, 456 \, 790 + 0 \, 987 \, 654 \, 321\) | \(=\) | \(\ds 1 \, 111 \, 111 \, 111\) |
$22 \, 222 \, 222 \, 222 \, 222$ is Kaprekar
\(\ds 22 \, 222 \, 222 \, 222 \, 222^2\) | \(=\) | \(\ds 493 \, 827 \, 160 \, 493 \, 817 \, 283 \, 950 \, 617 \, 284\) | ||||||||||||
\(\ds 4 \, 938 \, 271 \, 604 \, 938 + 17 \, 283 \, 950 \, 617 \, 284\) | \(=\) | \(\ds 22 \, 222 \, 222 \, 222 \, 222\) |
Also see
- Definition:Kaprekar Triple
- Results about Kaprekar numbers can be found here.
Source of Name
This entry was named for Dattathreya Ramchandra Kaprekar.
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $9$
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $297$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $9$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $297$
- Weisstein, Eric W. "Kaprekar Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KaprekarNumber.html