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. The numbers are named after D. R. Kaprekar.

Definition and properties

Let $n$ be a natural number. We define the Kaprekar function for base $b>1$ and power $p>0$ $F_{p,b}:\mathbb {N} \rightarrow \mathbb {N}$ to be the following:

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

,

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

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

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

For example, in base 10, 45 is a 2-Kaprekar number, because

\beta =n^{2}{\bmod {b}}^{p}=45^{2}{\bmod {1}}0^{2}=25

\alpha ={\frac {n^{2}-\beta }{b^{p}}}={\frac {45^{2}-25}{10^{2}}}=20

F_{2,10}(45)=\alpha +\beta =20+25=45

A natural number $n$ is a sociable Kaprekar number if it is a periodic point for $F_{p,b}$ , where $F_{p,b}^{k}(n)=n$ for a positive integer $k$ , 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 $F_{p,b}^{i}(n)$ 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

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

and $\beta =m^{2}$ , $\alpha =b^{p}+2m$ , and $F_{p,b}(n)=b^{p}+2m+m^{2}=n+(m^{2}+m)>n$ . 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 $2^{n}(2^{n+1}-1)$ or $2^{n}(2^{n+1}+1)$ for natural number $n$ are Kaprekar numbers in base 2.

Set-theoric definition and unitary divisors

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

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[1] that there is a bijection between the unitary divisors of $N-1$ and the set $K(N)$ defined above. Let ${\text{Inv}}(a,c)$ 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 $e={\frac {N-1}{d}}$ and $\zeta (d)=d\ {\text{Inv}}(d,e)$ . 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 $X=d\ {\text{Inv}}(d,e)$ 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 $e={\frac {N-1}{d}}$ , and if $X=d\ {\text{Inv}}(d,e)$ then $N-X=e\ {\text{Inv}}(e,d)$ .

Kaprekar numbers for $F_{p,b}$

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

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

$X_{1}={\frac {b^{p}-1}{2}}=(2k+1)\sum _{i=0}^{p-1}b^{i}$ is a Kaprekar number.

Proof

Let

${\begin{aligned}X_{1}&={\frac {b^{p}-1}{2}}\\&={\frac {b-1}{2}}\sum _{i=0}^{p-1}b^{i}\\&={\frac {4k+3-1}{2}}\sum _{i=0}^{2n+1-1}b^{i}\\&=(2k+1)\sum _{i=0}^{2n}b^{i}\\\end{aligned}}$

Then,

${\begin{aligned}X_{1}^{2}&=\left({\frac {b^{p}-1}{2}}\right)^{2}\\&={\frac {b^{2p}-2b^{p}+1}{4}}\\&={\frac {b^{p}(b^{p}-2)+1}{4}}\\&={\frac {(4k+3)^{2n+1}(b^{p}-2)+1}{4}}\\&={\frac {(4k+3)^{2n}(b^{p}-2)(4k+4)-(4k+3)^{2n}(b^{p}-2)+1}{4}}\\&={\frac {-(4k+3)^{2n}(b^{p}-2)+1}{4}}+(k+1)(4k+3)^{2n}(b^{p}-2)\\&={\frac {-(4k+3)^{2n-1}(b^{p}-2)(4k+4)+(4k+3)^{2n-1}(b^{p}-2)+1}{4}}+(k+1)b^{2n}(b^{2n+1}-2)\\&={\frac {(4k+3)^{2n-1}(b^{p}-2)+1}{4}}+(k+1)b^{2n}(b^{p}-2)-(k+1)b^{2n-1}(b^{2n+1}-2)\\&={\frac {(4k+3)^{p-2}(b^{p}-2)+1}{4}}+\sum _{i=p-2}^{p-1}(-1)^{i}(k+1)b^{i}(b^{p}-2)\\&={\frac {(4k+3)^{p-2}(b^{p}-2)+1}{4}}+(b^{p}-2)(k+1)\sum _{i=p-2}^{p-1}(-1)^{i}b^{i}\\&={\frac {(4k+3)^{1}(b^{p}-2)+1}{4}}+(b^{p}-2)(k+1)\sum _{i=1}^{p-1}(-1)^{i}b^{i}\\&={\frac {-(b^{p}-2)+1}{4}}+(b^{p}-2)(k+1)\sum _{i=0}^{p-1}(-1)^{i}b^{i}\\&=(b^{p}-2)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)+{\frac {-b^{2n+1}+3}{4}}\\&=(b^{p}-2)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)+{\frac {-4b^{2n+1}+3b^{2n+1}+3}{4}}\\&=(b^{p}-2)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)-b^{p}+{\frac {3b^{2n+1}+3}{4}}\\&=(b^{p}-2)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)-b^{p}+{\frac {3(4k+3)^{p-2}+3}{4}}+3(k+1)\sum _{i=p-2}^{p-1}(-1)^{i}b^{i}\\&=(b^{p}-2)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)-b^{p}+{\frac {3(4k+3)^{1}+3}{4}}+3(k+1)\sum _{i=1}^{p-1}(-1)^{i}b^{i}\\&=(b^{p}-2)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)-b^{p}+{\frac {-3+3}{4}}+3(k+1)\sum _{i=0}^{p-1}(-1)^{i}b^{i}\\&=(b^{p}-2)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)+3(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)-b^{p}\\&=(b^{p}-2+3)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)-b^{p}\\&=(b^{p}+1)(k+1)\left(\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)-b^{p}\\&=(b^{p}+1)\left(-1+(k+1)\sum _{i=0}^{2n}(-1)^{i}b^{i}\right)+1\\&=(b^{p}+1)\left(k+(k+1)\sum _{i=1}^{2n}(-1)^{i}b^{i}\right)+1\\&=(b^{p}+1)\left(k+(k+1)\sum _{i=1}^{n}b^{2i}-b^{2i-1}\right)+1\\&=(b^{p}+1)\left(k+(k+1)\sum _{i=1}^{n}(b-1)b^{2i-1}\right)+1\\&=(b^{p}+1)\left(k+\sum _{i=1}^{n}((k+1)b-k-1)b^{2i-1}\right)+1\\&=(b^{p}+1)\left(k+\sum _{i=1}^{n}(kb+(4k+3)-k-1)b^{2i-1}\right)+1\\&=(b^{p}+1)\left(k+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)+1\\&=b^{p}\left(k+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)+\left(k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)\\\end{aligned}}$

The two numbers $\alpha$ and $\beta$ are

\beta =X_{1}^{2}{\bmod {b}}^{p}=k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}

\alpha ={\frac {X_{1}^{2}-\beta }{b^{p}}}=k+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}

and their sum is

${\begin{aligned}\alpha +\beta &=\left(k+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)+\left(k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)\\&=2k+1+\sum _{i=1}^{n}((2k)b+2(3k+2))b^{2i-1}\\&=2k+1+\sum _{i=1}^{n}((2k)b+(6k+4))b^{2i-1}\\&=2k+1+\sum _{i=1}^{n}((2k)b+(4k+3))b^{2i-1}+(2k+1)b^{2i-1}\\&=2k+1+\sum _{i=1}^{n}((2k+1)b)b^{2i-1}+(2k+1)b^{2i-1}\\&=2k+1+\sum _{i=1}^{n}(2k+1)b^{2i}+(2k+1)b^{2i-1}\\&=2k+1+\sum _{i=1}^{2n}(2k+1)b^{i}\\&=\sum _{i=0}^{2n}(2k+1)b^{i}\\&=(2k+1)\sum _{i=0}^{2n}b^{i}\\&=X_{1}\\\end{aligned}}$

Thus, $X_{1}$ is a Kaprekar number.

$X_{2}={\frac {b^{p}+1}{2}}=X_{1}+1$ is a Kaprekar number for all natural numbers $n$ .

Proof

Let

${\begin{aligned}X_{2}&={\frac {b^{2n+1}+1}{2}}\\&={\frac {b^{2n+1}-1}{2}}+1\\&=X_{1}+1\\\end{aligned}}$

Then,

${\begin{aligned}X_{2}^{2}&=(X_{1}+1)^{2}\\&=X_{1}^{2}+2X_{1}+1\\&=X_{1}^{2}+2X_{1}+1\\&=b^{p}\left(k+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)+\left(k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)+b^{p}-1+1\\&=b^{p}\left(k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)+\left(k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)\\\end{aligned}}$

The two numbers $\alpha$ and $\beta$ are

\beta =X_{2}^{2}{\bmod {b}}^{p}=k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}

\alpha ={\frac {X_{2}^{2}-\beta }{b^{p}}}=k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}

and their sum is

${\begin{aligned}\alpha +\beta &=\left(k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)+\left(k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}\right)\\&=2k+2+\sum _{i=1}^{n}((2k)b+2(3k+2))b^{2i-1}\\&=2k+2+\sum _{i=1}^{n}((2k)b+(6k+4))b^{2i-1}\\&=2k+2+\sum _{i=1}^{n}((2k)b+(4k+3))b^{2i-1}+(2k+1)b^{2i-1}\\&=2k+2+\sum _{i=1}^{n}((2k+1)b)b^{2i-1}+(2k+1)b^{2i-1}\\&=2k+2+\sum _{i=1}^{n}(2k+1)b^{2i}+(2k+1)b^{2i-1}\\&=2k+2+\sum _{i=1}^{2n}(2k+1)b^{i}\\&=1+\sum _{i=0}^{2n}(2k+1)b^{i}\\&=1+(2k+1)\sum _{i=0}^{2n}b^{i}\\&=1+X_{1}\\&=X_{2}\\\end{aligned}}$

Thus, $X_{2}$ is a Kaprekar number.

b = m²k + m + 1 and p = mn + 1

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

$X_{1}={\frac {b^{p}-1}{m}}=(mk+1)\sum _{i=0}^{p-1}b^{i}$ is a Kaprekar number.
$X_{2}={\frac {b^{p}+m-1}{m}}=X_{1}+1$ is a Kaprekar number.

b = m²k + m + 1 and p = mn + m - 1

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

$X_{1}={\frac {m(b^{p}-1)}{4}}=(m-1)(mk+1)\sum _{i=0}^{p-1}b^{i}$ is a Kaprekar number.
$X_{2}={\frac {mb^{p}+1}{4}}=X_{3}+1$ is a Kaprekar number.

b = m²k + m² - m + 1 and p = mn + 1

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

$X_{1}={\frac {(m-1)(b^{p}-1)}{m}}=(m-1)(mk+1)\sum _{i=0}^{p-1}b^{i}$ is a Kaprekar number.
$X_{2}={\frac {(m-1)b^{p}+1}{m}}=X_{1}+1$ is a Kaprekar number.

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

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

$X_{1}={\frac {b^{p}-1}{m}}=(mk+1)\sum _{i=0}^{p-1}b^{i}$ is a Kaprekar number.
$X_{2}={\frac {b^{p}+m-1}{m}}=X_{3}+1$ is a Kaprekar number.

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

All numbers are in base $b$ .

Base $b$	Power $p$	Nontrivial Kaprekar numbers $n\neq 0$ , $n\neq 1$	Cycles
2	1	10	$\varnothing$
3	1	2, 10	$\varnothing$
4	1	3, 10	$\varnothing$
5	1	4, 5, 10	$\varnothing$
6	1	5, 6, 10	$\varnothing$
7	1	3, 4, 6, 10	$\varnothing$
8	1	7, 10	2 → 4 → 2
9	1	8, 10	$\varnothing$
10	1	9, 10	$\varnothing$
11	1	5, 6, A, 10	$\varnothing$
12	1	B, 10	$\varnothing$
13	1	4, 9, C, 10	$\varnothing$
14	1	D, 10	$\varnothing$
15	1	7, 8, E, 10	2 → 4 → 2 9 → B → 9
16	1	6, A, F, 10	$\varnothing$
2	2	11	$\varnothing$
3	2	22, 100	$\varnothing$
4	2	12, 22, 33, 100	$\varnothing$
5	2	14, 31, 44, 100	$\varnothing$
6	2	23, 33, 55, 100	15 → 24 → 15 41 → 50 → 41
7	2	22, 45, 66, 100	$\varnothing$
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	$\varnothing$
3	4	121, 2102, 2222, 10000	$\varnothing$
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	$\varnothing$
3	8	2012021, 10121020, 12101210, 21121001, 20210202, 22222222, 100000000	$\varnothing$
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.

Programming exercise

The example below implements the Kaprekar function described in the definition above to search for Kaprekar numbers and cycles in Python.

def kaprekarf(x: int, p: int, b: int) -> int:
    beta = pow(x, 2) % pow(b, p)
    alpha = (pow(x, 2) - beta) // pow(b, p)
    y = alpha + beta
    return y

def kaprekarf_cycle(x: int, p: int, b: int) -> List[int]:
    seen = []
    while x < pow(b, p) and x not in seen:
        seen.append(x)
        x = kaprekarf(x, p, b)
    if x > pow(b, p):
        return []
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = kaprekarf(x, p, b)
    return cycle

Notes

Iannucci (2000)

References

D. R. Kaprekar (1980–1981). "On Kaprekar numbers". Journal of Recreational Mathematics. 13: 81–82.
M. Charosh (1981–1982). "Some Applications of Casting Out 999...'s". Journal of Recreational Mathematics. 14: 111–118.
Iannucci, Douglas E. (2000). "The Kaprekar Numbers". Journal of Integer Sequences. 3: 00.1.2.CS1 maint: ref=harv (link)

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[iannucci-1] Iannucci (2000)

Kaprekar number

Definition and properties

Set-theoric definition and unitary divisors

Kaprekar numbers for F p , b {\displaystyle F_{p,b}}

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

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

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

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

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

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