In mathematics, a natural number in a given number base is a p-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p 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.

Definition and properties

Let n be a natural number. Then the Kaprekar function for base b > 1 and power p > 0 is defined to be the following: where and A natural number n is a p-Kaprekar number if it is a fixed point for F_{p, b}, which occurs if. 0 and 1 are trivial Kaprekar numbers for all b and p, all other Kaprekar numbers are nontrivial Kaprekar numbers. The earlier example of 45 satisfies this definition with b = 10 and p = 2, because A natural number n is a sociable Kaprekar number if it is a periodic point for F_{p, b}, where for a positive integer k (where F_{p, b}^k is the kth iterate of F_{p, b}), and forms a cycle of period k. A Kaprekar number is a sociable Kaprekar number with k = 1, and a amicable Kaprekar number is a sociable Kaprekar number with k = 2. The number of iterations i needed for to reach a fixed point is the Kaprekar function's persistence of n, and undefined if it never reaches a fixed point. There are only a finite number of p-Kaprekar numbers and cycles for a given base b, because if n = b^p + m, where m > 0 then and \beta = m^2,, and. Only when n \leq b^p do Kaprekar numbers and cycles exist. If d is any divisor of p, then n is also a p-Kaprekar number for base b^p. In base b = 2, all even perfect numbers are Kaprekar numbers. More generally, any numbers of the form or for natural number n are Kaprekar numbers in base 2.

Set-theoretic definition and unitary divisors

The set K(N) for a given integer N can be defined as the set of integers X for which there exist natural numbers A and B satisfying the Diophantine equation An n-Kaprekar number for base b is then one which lies in the set K(b^n). It was shown in 2000 that there is a bijection between the unitary divisors of N - 1 and the set K(N) defined above. Let denote the multiplicative inverse of a modulo c, namely the least positive integer m such that, and for each unitary divisor d of N - 1 let and. Then the function \zeta is a bijection from the set of unitary divisors of N - 1 onto the set K(N). In particular, a number X is in the set K(N) if and only if for some unitary divisor d of N - 1. The numbers in K(N) occur in complementary pairs, X and N - X. If d is a unitary divisor of N - 1 then so is, and if then.

Kaprekar numbers for F_{p, b}

b = 4k + 3 and p = 2n + 1

Let k and n be natural numbers, the number base, and p = 2n + 1. Then:

b = m2k + m + 1 and p = mn + 1

Let m, k, and n be natural numbers, the number base, and the power p = mn + 1. Then:

b = m2k + m + 1 and p = mn + m − 1

Let m, k, and n be natural numbers, the number base, and the power. Then:

b = m2k + m2 − m + 1 and p = mn + 1

Let m, k, and n be natural numbers, the number base, and the power. Then:

b = m2k + m2 − m + 1 and p = mn + m − 1

Let m, k, and n be natural numbers, the number base, and the power. Then:

Kaprekar numbers and cycles of F_{p, b} for specific p, b

All numbers are in base b.

Extension to negative integers

Kaprekar numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.