# 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

$$\sum _{i=\frac{k}{2}+1}^{k}{d}_{i}{b}^{i-\frac{k}{2}-1}+\sum _{i=1}^{\frac{k}{2}}{d}_{i}{b}^{i-1}=n$$ |

if $k$ is even or

$$\sum _{i=\lceil \frac{k}{2}\rceil}^{k}{d}_{i}{b}^{i-\lfloor \frac{k}{2}\rfloor -1}+\sum _{i=1}^{\frac{k}{2}}{d}_{i}{b}^{i-1}=n$$ |

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 |
---|---|

Canonical name | KaprekarNumber |

Date of creation | 2013-03-22 16:00:17 |

Last modified on | 2013-03-22 16:00:17 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 7 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A63 |