In mathematics, a natural number in a given number base is a
-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has
digits, that add up to the original number.
For example, in base 10, 45 is a 2-Kaprekar number, because 45² = 2025, and 20 + 25 = 45.
The numbers are named after D. R. Kaprekar.
Then the Kaprekar function for base
-Kaprekar number if it is a fixed point for
are trivial Kaprekar numbers for all
, all other Kaprekar numbers are nontrivial Kaprekar numbers.
The earlier example of 45 satisfies this definition with
is a sociable Kaprekar number if it is a periodic point for
), and forms a cycle of period
A Kaprekar number is a sociable Kaprekar number with
, and a amicable Kaprekar number is a sociable Kaprekar number with
to reach a fixed point is the Kaprekar function's persistence of
, and undefined if it never reaches a fixed point.
-Kaprekar numbers and cycles for a given base
do Kaprekar numbers and cycles exist.
More generally, any numbers of the form
are Kaprekar numbers in base 2.
can be defined as the set of integers
for which there exist natural numbers
satisfying the Diophantine equation[1] An
It was shown in 2000[1] that there is a bijection between the unitary divisors of
denote the multiplicative inverse of
, namely the least positive integer
is a bijection from the set of unitary divisors of
occur in complementary pairs,
α + β
15 → 24 → 15 41 → 50 → 41 4 → 20 → 4 11 → 22 → 11 45 → 56 → 45 10 → 100 → 10000 → 1000 → 10 111 → 10010 → 1110 → 1010 → 111 100 → 10000 → 100 1001 → 10010 → 1001 100101 → 101110 → 100101 100 → 10000 → 100 122012 → 201212 → 122012 10 → 100 → 10000 → 10 1000 → 1000000 → 100000 → 1000 100110 → 101111 → 110010 → 1010111 → 1001100 → 111101 → 100110 10 → 100 → 10000 → 10 1000 → 1000000 → 100000 → 1000 1111121 → 1111211 → 1121111 → 1111121 10 → 100 → 10000 → 100000000 → 10000000 → 100000 → 10 1000 → 1000000 → 1000 10011010 → 11010010 → 10011010 Kaprekar numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.