Kaprekar number

Short description: Base-dependent property of integers

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

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

where $\beta =n^2{modb}^p$ 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.

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

$\beta =n^2{modb}^p=45^{2}mod10^2=25$

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

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

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 $k$ (where $F_{p,b}^k$ is the $k$th 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 $F_{p,b}^i(n)$ 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

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

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 $n\le b^p$ 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-theoretic 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

$X^2=AN+B$, where $0\le B<N$

$X=A+B$

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 $\mathrm{Inv}(a,c)$ denote the multiplicative inverse of $a$ modulo $c$, namely the least positive integer $m$ such that $am=1modc$, 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 $X=d\mathrm{Inv}(d,e)$ then $N-X=e\mathrm{Inv}(e,d)$.

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

$\begin{array}{rl}{X}_1& =\frac{b^p-1}{2}\\ & =\frac{b-1}{2}\sum _{i=0}^{p-1}{b}^i\\ & =\frac{4k+3-1}{2}\sum _{i=0}^{2n+1-1}{b}^i\\ & =(2k+1)\sum _{i=0}^{2n}{b}^i\end{array}$

Then,

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

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

$\beta =X_1^2{modb}^p=k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1}$

$\alpha =\frac{X_1^2-\beta }{b^p}=k+\sum _{i=1}^n(kb+(3k+2))b^{2i-1}$

and their sum is

$\begin{array}{rl}\alpha +\beta & =(k+\sum _{i=1}^n(kb+(3k+2))b^{2i-1})+(k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1})\\ & =2k+1+\sum _{i=1}^n((2k)b+2(3k+2))b^{2i-1}\\ & =2k+1+\sum _{i=1}^n((2k)b+(6k+4))b^{2i-1}\\ & =2k+1+\sum _{i=1}^n((2k)b+(4k+3))b^{2i-1}+(2k+1)b^{2i-1}\\ & =2k+1+\sum _{i=1}^n((2k+1)b)b^{2i-1}+(2k+1)b^{2i-1}\\ & =2k+1+\sum _{i=1}^n(2k+1)b^{2i}+(2k+1)b^{2i-1}\\ & =2k+1+\sum _{i=1}^{2n}(2k+1)b^i\\ & =\sum _{i=0}^{2n}(2k+1)b^i\\ & =(2k+1)\sum _{i=0}^{2n}{b}^i& =X_1\\ \end{array}$

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

$\begin{array}{rl}{X}_2& =\frac{b^{2n+1}+1}{2}\\ & =\frac{b^{2n+1}-1}{2}+1\\ & =X_1+1\end{array}$

Then,

$\begin{array}{rl}{X}_2^2& =(X_1+1{)}^2\\ & =X_1^2+2X_1+1\\ & =X_1^2+2X_1+1\\ & =b^p(k+\sum _{i=1}^n(kb+(3k+2))b^{2i-1})+(k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1})+b^p-1+1\\ & =b^p(k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1})+(k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1})\end{array}$

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

$\beta =X_2^2{modb}^p=k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1}$

$\alpha =\frac{X_2^2-\beta }{b^p}=k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1}$

and their sum is

$\begin{array}{rl}\alpha +\beta & =(k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1})+(k+1+\sum _{i=1}^n(kb+(3k+2))b^{2i-1})\\ & =2k+2+\sum _{i=1}^n((2k)b+2(3k+2))b^{2i-1}\\ & =2k+2+\sum _{i=1}^n((2k)b+(6k+4))b^{2i-1}\\ & =2k+2+\sum _{i=1}^n((2k)b+(4k+3))b^{2i-1}+(2k+1)b^{2i-1}\\ & =2k+2+\sum _{i=1}^n((2k+1)b)b^{2i-1}+(2k+1)b^{2i-1}\\ & =2k+2+\sum _{i=1}^n(2k+1)b^{2i}+(2k+1)b^{2i-1}\\ & =2k+2+\sum _{i=1}^{2n}(2k+1)b^i\\ & =1+\sum _{i=0}^{2n}(2k+1)b^i\\ & =1+(2k+1)\sum _{i=0}^{2n}{b}^i\\ & =1+X_1\\ & =X_2\end{array}$

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.
• $X_2=\frac{m{b}^p+1}{4}=X_3+1$ 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 $n\ne 0$, $n\ne 1$	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.

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

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. Bibcode: 2000JIntS...3...12I. https://cs.uwaterloo.ca/journals/JIS/VOL3/iann2a.html.