Kaprekar Number

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 $p > 0$ $F_ : \mathbb \rightarrow \mathbb$ is defined to be the following:
:$F_(n) = \alpha + \beta$,
where $\beta = n^2 \bmod b^p$ and
:$\alpha = \frac$
A natural number $n$ is a $p$-Kaprekar number if it is a fixed point for $F_$, which occurs if $F_(n) = n$. $0$ 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
: $\beta = n^2 \bmod b^p = 45^2 \bmod 10^2 = 25$
: $\alpha = \frac = \frac = 20$
: $F_(45) = \alpha + \beta = 20 + 25 = 45$

A natural number $n$ is a sociable Kaprekar number if it is a periodic point for $F_$, where $F_^k(n) = n$ for a positive integer $k$ (where $F_^k$ is the $k$th iterate of $F_$), 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 $F_^(n)$ 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

: $\begin n^2 & = (b^p + m)^2 \\ & = b^ + 2mb^p + m^2 \\ & = (b^p + 2m)b^p + m^2 \\ \end$

and $\beta = m^2$, $\alpha = b^p + 2m$, and $F_(n) = b^p + 2m + m^2 = n + (m^2 + m) > n$. 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 $2^n (2^ - 1)$ or $2^n (2^ + 1)$ 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 equationIannucci (2000)
: $X^2 = AN + B$, where $0 \leq B < N$
: $X = A + B$
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 $\operatorname(a, c)$ denote the multiplicative inverse of $a$ modulo $c$, namely the least positive integer $m$ such that $am = 1 \bmod c$, and for each unitary divisor $d$ of $N - 1$ let $e = \frac$ and $\zeta(d) = d\ \text(d, e)$. 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 $X = d\ \text(d, e)$ 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 $e = \frac$, and if $X = d\operatorname(d, e)$ then $N - X = e\operatorname(e, d)$.

Kaprekar numbers for $F_$

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

Let $k$ and $n$ be natural numbers, the number base $b = 4k + 3 = 2(2k + 1) + 1$, and $p = 2n + 1$. Then:
* $X_1 = \frac = (2k + 1) \sum_^ b^i$ is a Kaprekar number.

* $X_2 = \frac = X_1 + 1$ is a Kaprekar number for all natural numbers $n$.

''b'' = ''m''²''k'' + ''m'' + 1 and ''p'' = ''mn'' + 1

Let $m$, $k$, and $n$ be natural numbers, the number base $b = m^2k + m + 1$, and the power $p = mn + 1$. Then:
* $X_1 = \frac = (mk + 1) \sum_^ b^i$ is a Kaprekar number.
* $X_2 = \frac = X_1 + 1$ is a Kaprekar number.

''b'' = ''m''²''k'' + ''m'' + 1 and ''p'' = ''mn'' + ''m'' − 1

Let $m$, $k$, and $n$ be natural numbers, the number base $b = m^2k + m + 1$, and the power $p = mn + m - 1$. Then:
* $X_1 = \frac = (m - 1)(mk + 1) \sum_^ b^i$ is a Kaprekar number.
* $X_2 = \frac = X_3 + 1$ is a Kaprekar number.

''b'' = ''m''²''k'' + ''m''² − ''m'' + 1 and ''p'' = ''mn'' + 1

Let $m$, $k$, and $n$ be natural numbers, the number base $b = m^2k + m^2 - m + 1$, and the power $p = mn + m - 1$. Then:
* $X_1 = \frac = (m - 1)(mk + 1) \sum_^ b^i$ is a Kaprekar number.
* $X_2 = \frac = X_1 + 1$ is a Kaprekar number.

''b'' = ''m''²''k'' + ''m''² − ''m'' + 1 and ''p'' = ''mn'' + ''m'' − 1

Let $m$, $k$, and $n$ be natural numbers, the number base $b = m^2k + m^2 - m + 1$, and the power $p = mn + m - 1$. Then:
* $X_1 = \frac = (mk + 1) \sum_^ b^i$ is a Kaprekar number.
* $X_2 = \frac = X_3 + 1$ is a Kaprekar number.

Kaprekar numbers and cycles of $F_$ 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.

See also

* Arithmetic dynamics
* Automorphic number
* Dudeney number
* Factorion
* Happy number
* Kaprekar's constant
* Meertens number
* Narcissistic number
* Perfect digit-to-digit invariant
* Perfect digital invariant
* Sum-product number

Notes

References

*
*
*

{{Classes of natural numbers

Arithmetic dynamics
Base-dependent integer sequences
Diophantine equations
Eponymous numbers in mathematics
Number theory
Indian inventions