Download PDF

Kaprekar number∗
PrimeFan†
2013-03-21 20:43:08
Let n be a k-digit integer in base b. Then n is said to be a Kaprekar number
in base b if n2 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
n2 =
k−1
X
di bi
i=0
(where dx are digits, with d0 the least significant digit and dk−1 the most significant) such that
k
k
X
di b
i− k
2 −1
+
2
X
di bi−1 = n
i=1
i= k
2 +1
if k is even or
k
X
k
di b
i−b k
2 c−1
+
2
X
di bi−1 = n
i=1
i=d k
2e
if k is odd.
bx − 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.
∗ hKaprekarNumberi
created: h2013-03-21i by: hPrimeFani version: h38034i Privacy
setting: h1i hDefinitioni h11A63i
† This text is available under the Creative Commons Attribution/Share-Alike License 3.0.
You can reuse this document or portions thereof only if you do so under terms that are
compatible with the CC-BY-SA license.
1