Inmathematics, anatural number in a givennumber 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, inbase 10, 45 is a 2-Kaprekar number, because 45² = 2025, and 20 + 25 = 45. The numbers are named afterD. R. Kaprekar.

Let${\displaystyle n}$ be a natural number. Then theKaprekar function for base${\displaystyle b>1}$ and power${\displaystyle p>0}$${\displaystyle F_{p,b}:\mathbb{N}\to \mathbb{N}}$ is defined to be the following:

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

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

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

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

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

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

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

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

A natural number${\displaystyle n}$ is asociable Kaprekar number if it is aperiodic point for${\displaystyle F_{p,b}}$, where${\displaystyle F_{p,b}^k(n)=n}$ for a positiveinteger${\displaystyle k}$ (where${\displaystyle F_{p,b}^k}$ is the${\displaystyle k}$thiterate of${\displaystyle F_{p,b}}$), and forms acycle of period${\displaystyle k}$. A Kaprekar number is a sociable Kaprekar number with${\displaystyle k=1}$, and aamicable 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'spersistence 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${\displaystyle m>0}$ then

and${\displaystyle \beta =m^2}$,${\displaystyle \alpha =b^p+2m}$, and${\displaystyle F_{p,b}(n)=b^p+2m+m^2=n+(m^2+m)>n}$. Only when${\displaystyle n\le b^p}$ 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 evenperfect 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 inbase 2.

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

${\displaystyle X^2=AN+B}$, where

${\displaystyle X=A+B}$

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 abijection between theunitary divisors of${\displaystyle N-1}$ and the set${\displaystyle K(N)}$ defined above. Let${\displaystyle \mathrm{Inv}(a,c)}$ denote themultiplicative inverse of${\displaystyle a}$ modulo${\displaystyle c}$, namely the least positive integer${\displaystyle m}$ such that${\displaystyle am=1modc}$, 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${\displaystyle X=d\mathrm{Inv}(d,e)}$ then${\displaystyle N-X=e\mathrm{Inv}(e,d)}$.

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${\displaystyle n}$.

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

Let${\displaystyle m}$,${\displaystyle k}$, and${\displaystyle n}$ be natural numbers, the number base${\displaystyle b=m^{2}k+m+1}$, 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.
• ${\displaystyle X_2=\frac{m{b}^p+1}{4}=X_3+1}$ is a Kaprekar number.

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

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

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

Base${\displaystyle b}$	Power${\displaystyle p}$	Nontrivial Kaprekar numbers${\displaystyle n\ne 0}$,${\displaystyle n\ne 1}$	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

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

• 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

• 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.

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

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