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Kaprekar number (Definition)

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

$\displaystyle n^2 = \sum_{i = 0}^{k - 1} d_ib^i$
(where $ d_x$ are digits, with $ d_0$ the least significant digit and $ d_{k - 1}$ the most significant) such that
$\displaystyle \sum_{i = {k \over 2} + 1}^k d_ib^{i - {k \over 2} - 1} + \sum_{i = 1}^{k \over 2} d_ib^{i - 1} = n$
if $ k$ is even or
$\displaystyle \sum_{i = \lceil {k \over 2} \rceil}^k d_ib^{i - \lfloor {k \over 2} \rfloor - 1} + \sum_{i = 1}^{k \over 2} d_ib^{i - 1} = n$
if $ k$ is odd.

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

Bibliography

1
D. R. Kaprekar, ``On Kaprekar numbers" J. Rec. Math. 13 (1980-1981), 81 - 82.



"Kaprekar number" is owned by PrimeFan. [ full author list (2) | owner history (1) ]
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Cross-references: odd, even, least significant digit, digits, right, number, property, base, integer
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This is version 4 of Kaprekar number, born on 2006-06-14, modified 2007-07-22.
Object id is 8034, canonical name is KaprekarNumber.
Accessed 1020 times total.

Classification:
AMS MSC: 11A63 (Number theory :: Elementary number theory :: Radix representation; digital problems)
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