Let be a natural number. We define the Kaprekar function for base and power to be the following:
- ,
where and
A natural number is a -Kaprekar number if it is a fixed point for , which occurs if . and are trivial Kaprekar numbers for all and , all other Kaprekar numbers are nontrivial Kaprekar numbers.
The earlier example of 45 satisfies this definition with and , because
A natural number is a sociable Kaprekar number if it is a periodic point for , where for a positive integer (where is the th iterate of ), 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 .
The number of iterations needed for to reach a fixed point is the Kaprekar function's persistence of , and undefined if it never reaches a fixed point.
There are only a finite number of -Kaprekar numbers and cycles for a given base , because if , where then
and , , and . Only when do Kaprekar numbers and cycles exist.
If is any divisor of , then is also a -Kaprekar number for base .
In base , all even perfect numbers are Kaprekar numbers. More generally, any numbers of the form or for natural number are Kaprekar numbers in base 2.
Set-theoretic definition and unitary divisors
We can define the set for a given integer as the set of integers for which there exist natural numbers and satisfying the Diophantine equation[1]
- , where
An -Kaprekar number for base is then one which lies in the set .
It was shown in 2000[1] that there is a bijection between the unitary divisors of and the set defined above. Let denote the multiplicative inverse of modulo , namely the least positive integer such that , and for each unitary divisor of let and . Then the function is a bijection from the set of unitary divisors of onto the set . In particular, a number is in the set if and only if for some unitary divisor of .
The numbers in occur in complementary pairs, and . If is a unitary divisor of then so is , and if then .
b = 4k + 3 and p = 2n + 1
Let and be natural numbers, the number base , and . Then:
- is a Kaprekar number.
Proof
Let
Then,
The two numbers and are
and their sum is
Thus, is a Kaprekar number.
- is a Kaprekar number for all natural numbers .
Proof
Let
Then,
The two numbers and are
and their sum is
Thus, is a Kaprekar number.