Definition:Kaprekar's Process

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Definition

Kaprekar's process is the repeated application of the Kaprekar mapping to a given positive integer.


Examples

Kaprekar's Process on $4527$

From $4527$:

\(\displaystyle 7542 - 2457\) \(=\) \(\displaystyle 5085\)
\(\displaystyle 8550 - 0558\) \(=\) \(\displaystyle 7992\)
\(\displaystyle 9972 - 2799\) \(=\) \(\displaystyle 7173\)
\(\displaystyle 7731 - 1377\) \(=\) \(\displaystyle 6354\)
\(\displaystyle 6543 - 3456\) \(=\) \(\displaystyle 3087\)
\(\displaystyle 8730 - 0378\) \(=\) \(\displaystyle 8352\)
\(\displaystyle 8532 - 2358\) \(=\) \(\displaystyle 6174\)
\(\displaystyle 7641 - 1467\) \(=\) \(\displaystyle 6174\)

$\blacksquare$


Also known as

Kaprekar's process is also known as the Kaprekar routine or the Kaprekar sequence.

Sometimes rendered as the Kaprekar process.


Also defined as

Some sources define the Kaprekar mapping so as not to retain the leading zeroes, and so, for example:

$K \left({1121}\right) = 2111 - 1112 = 999$
$K \left({999}\right) = 999 - 999 = 0$

instead of:

$K \left({1121}\right) = 2111 - 1112 = 0999$
$K \left({0999}\right) = 9990 - 0999 = 8991$

The process as initially specified does retain all leading zeroes.


Also see


Source of Name

This entry was named for Dattathreya Ramchandra Kaprekar.


Sources