# Kaprekar number

Short description: Base-dependent property of integers

In mathematics, a natural number in a given number base is a $\displaystyle{ p }$-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has $\displaystyle{ p }$ 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 $\displaystyle{ n }$ be a natural number. We define the Kaprekar function for base $\displaystyle{ b \gt 1 }$ and power $\displaystyle{ p \gt 0 }$ $\displaystyle{ F_{p, b} : \mathbb{N} \rightarrow \mathbb{N} }$ to be the following:

$\displaystyle{ F_{p, b}(n) = \alpha + \beta }$,

where $\displaystyle{ \beta = n^2 \bmod b^p }$ and

$\displaystyle{ \alpha = \frac{n^2 - \beta}{b^p} }$

A natural number $\displaystyle{ n }$ is a $\displaystyle{ p }$-Kaprekar number if it is a fixed point for $\displaystyle{ F_{p, b} }$, which occurs if $\displaystyle{ F_{p, b}(n) = n }$. $\displaystyle{ 0 }$ and $\displaystyle{ 1 }$ are trivial Kaprekar numbers for all $\displaystyle{ b }$ and $\displaystyle{ p }$, all other Kaprekar numbers are nontrivial Kaprekar numbers.

The earlier example of 45 satisfies this definition with $\displaystyle{ b = 10 }$ and $\displaystyle{ p = 2 }$, because

$\displaystyle{ \beta = n^2 \bmod b^p = 45^2 \bmod 10^2 = 25 }$
$\displaystyle{ \alpha = \frac{n^2 - \beta}{b^p} = \frac{45^2 - 25}{10^2} = 20 }$
$\displaystyle{ F_{2, 10}(45) = \alpha + \beta = 20 + 25 = 45 }$

A natural number $\displaystyle{ n }$ is a sociable Kaprekar number if it is a periodic point for $\displaystyle{ F_{p, b} }$, where $\displaystyle{ F_{p, b}^k(n) = n }$ for a positive integer $\displaystyle{ k }$ (where $\displaystyle{ F_{p, b}^k }$ is the $\displaystyle{ k }$th iterate of $\displaystyle{ F_{p, b} }$), and forms a cycle of period $\displaystyle{ k }$. A Kaprekar number is a sociable Kaprekar number with $\displaystyle{ k = 1 }$, and a amicable Kaprekar number is a sociable Kaprekar number with $\displaystyle{ k = 2 }$.

The number of iterations $\displaystyle{ i }$ needed for $\displaystyle{ F_{p, b}^{i}(n) }$ to reach a fixed point is the Kaprekar function's persistence of $\displaystyle{ n }$, and undefined if it never reaches a fixed point.

There are only a finite number of $\displaystyle{ p }$-Kaprekar numbers and cycles for a given base $\displaystyle{ b }$, because if $\displaystyle{ n = b^p + m }$, where $\displaystyle{ m \gt 0 }$ then

\displaystyle{ \begin{align} n^2 & = (b^p + m)^2 \\ & = b^{2p} + 2mb^p + m^2 \\ & = (b^p + 2m)b^p + m^2 \\ \end{align} }

and $\displaystyle{ \beta = m^2 }$, $\displaystyle{ \alpha = b^p + 2m }$, and $\displaystyle{ F_{p, b}(n) = b^p + 2m + m^2 = n + (m^2 + m) \gt n }$. Only when $\displaystyle{ n \leq b^p }$ do Kaprekar numbers and cycles exist.

If $\displaystyle{ d }$ is any divisor of $\displaystyle{ p }$, then $\displaystyle{ n }$ is also a $\displaystyle{ p }$-Kaprekar number for base $\displaystyle{ b^p }$.

In base $\displaystyle{ b = 2 }$, all even perfect numbers are Kaprekar numbers. More generally, any numbers of the form $\displaystyle{ 2^n (2^{n + 1} - 1) }$ or $\displaystyle{ 2^n (2^{n + 1} + 1) }$ for natural number $\displaystyle{ n }$ are Kaprekar numbers in base 2.

### Set-theoretic definition and unitary divisors

We can define the set $\displaystyle{ K(N) }$ for a given integer $\displaystyle{ N }$ as the set of integers $\displaystyle{ X }$ for which there exist natural numbers $\displaystyle{ A }$ and $\displaystyle{ B }$ satisfying the Diophantine equation[1]

$\displaystyle{ X^2 = AN + B }$, where $\displaystyle{ 0 \leq B \lt N }$
$\displaystyle{ X = A + B }$

An $\displaystyle{ n }$-Kaprekar number for base $\displaystyle{ b }$ is then one which lies in the set $\displaystyle{ K(b^n) }$.

It was shown in 2000[1] that there is a bijection between the unitary divisors of $\displaystyle{ N - 1 }$ and the set $\displaystyle{ K(N) }$ defined above. Let $\displaystyle{ \operatorname{Inv}(a, c) }$ denote the multiplicative inverse of $\displaystyle{ a }$ modulo $\displaystyle{ c }$, namely the least positive integer $\displaystyle{ m }$ such that $\displaystyle{ am = 1 \bmod c }$, and for each unitary divisor $\displaystyle{ d }$ of $\displaystyle{ N - 1 }$ let $\displaystyle{ e = \frac{N - 1}{d} }$ and $\displaystyle{ \zeta(d) = d\ \text{Inv}(d, e) }$. Then the function $\displaystyle{ \zeta }$ is a bijection from the set of unitary divisors of $\displaystyle{ N - 1 }$ onto the set $\displaystyle{ K(N) }$. In particular, a number $\displaystyle{ X }$ is in the set $\displaystyle{ K(N) }$ if and only if $\displaystyle{ X = d\ \text{Inv}(d, e) }$ for some unitary divisor $\displaystyle{ d }$ of $\displaystyle{ N - 1 }$.

The numbers in $\displaystyle{ K(N) }$ occur in complementary pairs, $\displaystyle{ X }$ and $\displaystyle{ N - X }$. If $\displaystyle{ d }$ is a unitary divisor of $\displaystyle{ N - 1 }$ then so is $\displaystyle{ e = \frac{N - 1}{d} }$, and if $\displaystyle{ X = d\operatorname{Inv}(d, e) }$ then $\displaystyle{ N - X = e\operatorname{Inv}(e, d) }$.

## Kaprekar numbers for $\displaystyle{ F_{p, b} }$

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

Let $\displaystyle{ k }$ and $\displaystyle{ n }$ be natural numbers, the number base $\displaystyle{ b = 4k + 3 = 2(2k + 1) + 1 }$, and $\displaystyle{ p = 2n + 1 }$. Then:

• $\displaystyle{ X_1 = \frac{b^p - 1}{2} = (2k + 1) \sum_{i = 0}^{p - 1} b^i }$ is a Kaprekar number.
• $\displaystyle{ X_2 = \frac{b^p + 1}{2} = X_1 + 1 }$ is a Kaprekar number for all natural numbers $\displaystyle{ n }$.

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

Let $\displaystyle{ m }$, $\displaystyle{ k }$, and $\displaystyle{ n }$ be natural numbers, the number base $\displaystyle{ b = m^2k + m + 1 }$, and the power $\displaystyle{ p = mn + 1 }$. Then:

• $\displaystyle{ X_1 = \frac{b^p - 1}{m} = (mk + 1) \sum_{i = 0}^{p - 1} b^i }$ is a Kaprekar number.
• $\displaystyle{ X_2 = \frac{b^p + m - 1}{m} = X_1 + 1 }$ is a Kaprekar number.

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

Let $\displaystyle{ m }$, $\displaystyle{ k }$, and $\displaystyle{ n }$ be natural numbers, the number base $\displaystyle{ b = m^2k + m + 1 }$, and the power $\displaystyle{ p = mn + m - 1 }$. Then:

• $\displaystyle{ X_1 = \frac{m(b^p - 1)}{4} = (m - 1)(mk + 1) \sum_{i = 0}^{p - 1} b^i }$ is a Kaprekar number.
• $\displaystyle{ X_2 = \frac{mb^p + 1}{4} = X_3 + 1 }$ is a Kaprekar number.

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

Let $\displaystyle{ m }$, $\displaystyle{ k }$, and $\displaystyle{ n }$ be natural numbers, the number base $\displaystyle{ b = m^2k + m^2 - m + 1 }$, and the power $\displaystyle{ p = mn + m - 1 }$. Then:

• $\displaystyle{ X_1 = \frac{(m - 1)(b^p - 1)}{m} = (m - 1)(mk + 1) \sum_{i = 0}^{p - 1} b^i }$ is a Kaprekar number.
• $\displaystyle{ X_2 = \frac{(m - 1)b^p + 1}{m} = X_1 + 1 }$ is a Kaprekar number.

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

Let $\displaystyle{ m }$, $\displaystyle{ k }$, and $\displaystyle{ n }$ be natural numbers, the number base $\displaystyle{ b = m^2k + m^2 - m + 1 }$, and the power $\displaystyle{ p = mn + m - 1 }$. Then:

• $\displaystyle{ X_1 = \frac{b^p - 1}{m} = (mk + 1) \sum_{i = 0}^{p - 1} b^i }$ is a Kaprekar number.
• $\displaystyle{ X_2 = \frac{b^p + m - 1}{m} = X_3 + 1 }$ is a Kaprekar number.

## Kaprekar numbers and cycles of $\displaystyle{ F_{p, b} }$ for specific $\displaystyle{ p }$, $\displaystyle{ b }$

All numbers are in base $\displaystyle{ b }$.

Base $\displaystyle{ b }$ Power $\displaystyle{ p }$ Nontrivial Kaprekar numbers $\displaystyle{ n \neq 0 }$, $\displaystyle{ n \neq 1 }$ Cycles
2 1 10 $\displaystyle{ \varnothing }$
3 1 2, 10 $\displaystyle{ \varnothing }$
4 1 3, 10 $\displaystyle{ \varnothing }$
5 1 4, 5, 10 $\displaystyle{ \varnothing }$
6 1 5, 6, 10 $\displaystyle{ \varnothing }$
7 1 3, 4, 6, 10 $\displaystyle{ \varnothing }$
8 1 7, 10 2 → 4 → 2
9 1 8, 10 $\displaystyle{ \varnothing }$
10 1 9, 10 $\displaystyle{ \varnothing }$
11 1 5, 6, A, 10 $\displaystyle{ \varnothing }$
12 1 B, 10 $\displaystyle{ \varnothing }$
13 1 4, 9, C, 10 $\displaystyle{ \varnothing }$
14 1 D, 10 $\displaystyle{ \varnothing }$
15 1 7, 8, E, 10

2 → 4 → 2

9 → B → 9

16 1 6, A, F, 10 $\displaystyle{ \varnothing }$
2 2 11 $\displaystyle{ \varnothing }$
3 2 22, 100 $\displaystyle{ \varnothing }$
4 2 12, 22, 33, 100 $\displaystyle{ \varnothing }$
5 2 14, 31, 44, 100 $\displaystyle{ \varnothing }$
6 2 23, 33, 55, 100

15 → 24 → 15

41 → 50 → 41

7 2 22, 45, 66, 100 $\displaystyle{ \varnothing }$
8 2 34, 44, 77, 100

4 → 20 → 4

11 → 22 → 11

45 → 56 → 45

2 3 111, 1000 10 → 100 → 10
3 3 111, 112, 222, 1000 10 → 100 → 10
2 4 110, 1010, 1111, 10000 $\displaystyle{ \varnothing }$
3 4 121, 2102, 2222, 10000 $\displaystyle{ \varnothing }$
2 5 11111, 100000

10 → 100 → 10000 → 1000 → 10

111 → 10010 → 1110 → 1010 → 111

3 5 11111, 22222, 100000 10 → 100 → 10000 → 1000 → 10
2 6 11100, 100100, 111111, 1000000

100 → 10000 → 100

1001 → 10010 → 1001

100101 → 101110 → 100101

3 6 10220, 20021, 101010, 121220, 202202, 212010, 222222, 1000000

100 → 10000 → 100

122012 → 201212 → 122012

2 7 1111111, 10000000

10 → 100 → 10000 → 10

1000 → 1000000 → 100000 → 1000

100110 → 101111 → 110010 → 1010111 → 1001100 → 111101 → 100110

3 7 1111111, 1111112, 2222222, 10000000

10 → 100 → 10000 → 10

1000 → 1000000 → 100000 → 1000

1111121 → 1111211 → 1121111 → 1111121

2 8 1010101, 1111000, 10001000, 10101011, 11001101, 11111111, 100000000 $\displaystyle{ \varnothing }$
3 8 2012021, 10121020, 12101210, 21121001, 20210202, 22222222, 100000000 $\displaystyle{ \varnothing }$
2 9 10010011, 101101101, 111111111, 1000000000

10 → 100 → 10000 → 100000000 → 10000000 → 100000 → 10

1000 → 1000000 → 1000

10011010 → 11010010 → 10011010

## Extension to negative integers

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