# Kaprekar number

Let $n$ be a $k$-digit integer in base $b$. Then $n$ is said to be a Kaprekar number in base $b$ if $n^{2}$ has the following property: when you add the number formed by its right hand digits to that formed by its left hand digits, you get $n$.

Or to put it algebraically, an integer $n$ such that in a given base $b$ has

 $n^{2}=\sum_{i=0}^{k-1}d_{i}b^{i}$

(where $d_{x}$ are digits, with $d_{0}$ the least significant digit and $d_{k-1}$ the most significant) such that

 $\sum_{i={k\over 2}+1}^{k}d_{i}b^{i-{k\over 2}-1}+\sum_{i=1}^{k\over 2}d_{i}b^{% i-1}=n$

if $k$ is even or

 $\sum_{i=\lceil{k\over 2}\rceil}^{k}d_{i}b^{i-\lfloor{k\over 2}\rfloor-1}+\sum_% {i=1}^{k\over 2}d_{i}b^{i-1}=n$

if $k$ is odd.

$b^{x}-1$ for a natural $x$ is always a Kaprekar number in base $b$.

## References

• 1 D. R. Kaprekar, “On Kaprekar numbers” J. Rec. Math. 13 (1980-1981), 81 - 82.
Title Kaprekar number KaprekarNumber 2013-03-22 16:00:17 2013-03-22 16:00:17 PrimeFan (13766) PrimeFan (13766) 7 PrimeFan (13766) Definition msc 11A63