# Kaprekar constant

The Kaprekar constant $K_{k}$ in a given base $b$ is a $k$-digit number $K$ such that subjecting any other $k$-digit number $n$ (except the repunit $R_{k}$ and numbers with $k-1$ repeated digits) to the following process:

1. Arrange the digits of $n$ in ascending order, forming the $k$-digit number $a$, and then in descending order, forming the $k$-digit number $b$.

2. If $a>b$, calculate $a-b=c$; otherwise $b-a=c$.

3. Goto step 1 using $c$ instead of $n$.

eventually gives $K$. (This process is sometimes called the Kaprekar routine).

For $b=10$, the Kaprekar constant for $k=4$ is 6174. Using $n=1729$, we find that 9721 - 1279 gives 8442. Then 8442 - 2448 = 5994. Then 9954 - 4599 gives 5355. Then 5553 - 3555 gives 1998. Then 9981 - 1899 gives 8082. Then 8820 - 288 gives 8532. Then 8532 - 2538 finally gives 6174. (Some numbers take longer than others). $K_{2}$ and $K_{7}$ don’t exist for $b=10$.

Title Kaprekar constant KaprekarConstant 2013-03-22 16:16:30 2013-03-22 16:16:30 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Definition msc 11A63 Kaprekar’s constant Kaprekar routine