Kaprekar number – Wikipedia

Base-dependent property of integers

In mathematics, a natural number in a given number 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. The numbers are named after D. R. Kaprekar.

Definition and properties[edit]

Let

${displaystyle n}$

be a natural number. We define the Kaprekar function for base

${displaystyle b>1}$

${displaystyle p>0}$

${displaystyle F_{p,b}:mathbb {N} rightarrow mathbb {N} }$

to be the following:

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

,

where

${displaystyle beta =n^{2}{bmod {b}}^{p}}$

and

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

A natural number

${displaystyle n}$

is a

${displaystyle p}$

–Kaprekar number if it is a fixed point for

${displaystyle F_{p,b}}$

, which occurs if

${displaystyle F_{p,b}(n)=n}$

.

${displaystyle 0}$

and

${displaystyle 1}$

are trivial Kaprekar numbers for all

${displaystyle b}$

and

${displaystyle p}$

, all other Kaprekar numbers are nontrivial Kaprekar numbers.

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

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

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

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

A natural number

${displaystyle n}$

is a sociable Kaprekar number if it is a periodic point for

${displaystyle F_{p,b}}$

, where

${displaystyle F_{p,b}^{k}(n)=n}$

for a positive integer

${displaystyle k}$

(where

${displaystyle F_{p,b}^{k}}$

is the

${displaystyle k}$

th iterate of

${displaystyle F_{p,b}}$

), and forms a cycle of period

${displaystyle k}$

. A Kaprekar number is a sociable Kaprekar number with

${displaystyle k=1}$

, and a amicable 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’s persistence 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

${displaystyle m>0}$

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

and

${displaystyle beta =m^{2}}$

,

${displaystyle alpha =b^{p}+2m}$

, and

${displaystyle F_{p,b}(n)=b^{p}+2m+m^{2}=n+(m^{2}+m)>n}$

${displaystyle nleq b^{p}}$

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 even perfect 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 in base 2.

Set-theoretic definition and unitary divisors[edit]

We can define the set

${displaystyle K(N)}$

for a given integer

${displaystyle N}$

as the set of integers

${displaystyle X}$

for which there exist natural numbers

${displaystyle A}$

and

${displaystyle B}$

satisfying the Diophantine equation^[1]

${displaystyle X^{2}=AN+B}$

, where

${displaystyle 0leq B$

${displaystyle X=A+B}$

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 a bijection between the unitary divisors of

${displaystyle N-1}$

and the set

${displaystyle K(N)}$

defined above. Let

${displaystyle operatorname {Inv} (a,c)}$

denote the multiplicative inverse of

${displaystyle a}$

modulo

${displaystyle c}$

, namely the least positive integer

${displaystyle m}$

such that

${displaystyle am=1{bmod {c}}}$

, 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

${displaystyle X=doperatorname {Inv} (d,e)}$

then

${displaystyle N-X=eoperatorname {Inv} (e,d)}$

.

Kaprekar numbers for

${displaystyle F_{p,b}}$

[

$$



F

p
,
b




{displaystyle F_{p,b}}

“>edit]

b = 4k + 3 and p = 2n + 1[edit]

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

  ${displaystyle n}$

  .

b = m²k + m + 1 and p = mn + 1[edit]

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.

b = m²k + m + 1 and p = mn + m − 1[edit]

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.

• 
  ${displaystyle X_{2}={frac {mb^{p}+1}{4}}=X_{3}+1}$

  is a Kaprekar number.

b = m²k + m² − m + 1 and p = mn + 1[edit]

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.

b = m²k + m² − m + 1 and p = mn + m − 1[edit]

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.

Kaprekar numbers and cycles of

${displaystyle F_{p,b}}$

for specific

${displaystyle p}$

,

${displaystyle b}$

[

$$



F

p
,
b




{displaystyle F_{p,b}}

for specific

$$


p


{displaystyle p}

,

$$


b


{displaystyle b}

“>edit]

All numbers are in base

${displaystyle b}$

.

Base ${displaystyle b}$	Power ${displaystyle p}$	Nontrivial Kaprekar numbers ${displaystyle nneq 0}$ , ${displaystyle nneq 1}$	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

Extension to negative integers[edit]

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

See also[edit]

References[edit]