Kaprekar number

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 $p>0$ $F_{p,b}:\mathbb {N} \rightarrow \mathbb {N}$ 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.

The earlier example of 45 satisfies this definition with $b=10$ and $p=2$ , 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 $k$ (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{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-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^[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 $\operatorname {Inv} (a,c)$ 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 $X=d\operatorname {Inv} (d,e)$ then $N-X=e\operatorname {Inv} (e,d)$ .

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.

Notes

^ ^a ^b 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. Bibcode:2000JIntS...3...12I.

Classes of natural numbers

Powers and related numbers
Achilles Power of 2 Power of 3 Power of 10 Square Cube Fourth power Fifth power Sixth power Seventh power Eighth power Perfect power Powerful Prime power

Of the form a × 2^b ± 1
Cullen Double Mersenne Fermat Mersenne Proth Thabit Woodall

Other polynomial numbers
Hilbert Idoneal Leyland Loeschian Lucky numbers of Euler

Recursively defined numbers
Fibonacci Jacobsthal Leonardo Lucas Narayana Padovan Pell Perrin

Possessing a specific set of other numbers
Amenable Congruent Knödel Riesel Sierpiński

Expressible via specific sums
Nonhypotenuse Polite Practical Primary pseudoperfect Ulam Wolstenholme

Figurate numbers

2-dimensional

centered	Centered triangular Centered square Centered pentagonal Centered hexagonal Centered heptagonal Centered octagonal Centered nonagonal Centered decagonal Star
non-centered	Triangular Square Square triangular Pentagonal Hexagonal Heptagonal Octagonal Nonagonal Decagonal Dodecagonal

3-dimensional

centered	Centered tetrahedral Centered cube Centered octahedral Centered dodecahedral Centered icosahedral
non-centered	Tetrahedral Cubic Octahedral Dodecahedral Icosahedral Stella octangula
pyramidal	Square pyramidal

4-dimensional

non-centered	Pentatope Squared triangular Tesseractic

Combinatorial numbers
Bell Cake Catalan Dedekind Delannoy Euler Eulerian Fuss–Catalan Lah Lazy caterer's sequence Lobb Motzkin Narayana Ordered Bell Schröder Schröder–Hipparchus Stirling first Stirling second Telephone number Wedderburn–Etherington

Primes
Wieferich Wall–Sun–Sun Wolstenholme prime Wilson

Pseudoprimes
Carmichael number Catalan pseudoprime Elliptic pseudoprime Euler pseudoprime Euler–Jacobi pseudoprime Fermat pseudoprime Frobenius pseudoprime Lucas pseudoprime Lucas–Carmichael number Perrin pseudoprime Somer–Lucas pseudoprime Strong pseudoprime

Arithmetic functions and dynamics

Divisor functions	Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical Primitive abundant Quasiperfect Refactorable Semiperfect Sublime Superabundant Superior highly composite Superperfect
Prime omega functions	Almost prime Semiprime
Euler's totient function	Highly cototient Highly totient Noncototient Nontotient Perfect totient Sparsely totient
Aliquot sequences	Amicable Perfect Sociable Untouchable
Primorial	Euclid Fortunate

Other prime factor or divisor related numbers
Blum Cyclic Erdős–Nicolas Erdős–Woods Friendly Giuga Harmonic divisor Jordan–Pólya Lucas–Carmichael Pronic Regular Rough Smooth Sphenic Størmer Super-Poulet Zeisel

Numeral system-dependent numbers

Arithmetic functions
and dynamics

Persistence
- Additive
- Multiplicative

Digit sum	Digit sum Digital root Self Sum-product
Digit product	Multiplicative digital root Sum-product
Coding-related	Meertens
Other	Dudeney Factorion Kaprekar Kaprekar's constant Keith Lychrel Narcissistic Perfect digit-to-digit invariant Perfect digital invariant Happy