Kaprekar number

Summary

In mathematics, a natural number in a given number base is a -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has 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 edit

Let   be a natural number. We define the Kaprekar function for base   and power     to be the following:

 ,

where   and

 

A natural number   is a  -Kaprekar number if it is a fixed point for  , which occurs if  .   and   are trivial Kaprekar numbers for all   and  , all other Kaprekar numbers are nontrivial Kaprekar numbers.

The earlier example of 45 satisfies this definition with   and  , because

 
 
 

A natural number   is a sociable Kaprekar number if it is a periodic point for  , where   for a positive integer   (where   is the  th iterate of  ), and forms a cycle of period  . A Kaprekar number is a sociable Kaprekar number with  , and a amicable Kaprekar number is a sociable Kaprekar number with  .

The number of iterations   needed for   to reach a fixed point is the Kaprekar function's persistence of  , and undefined if it never reaches a fixed point.

There are only a finite number of  -Kaprekar numbers and cycles for a given base  , because if  , where   then

 

and  ,  , and  . Only when   do Kaprekar numbers and cycles exist.

If   is any divisor of  , then   is also a  -Kaprekar number for base  .

In base  , all even perfect numbers are Kaprekar numbers. More generally, any numbers of the form   or   for natural number   are Kaprekar numbers in base 2.

Set-theoretic definition and unitary divisors edit

We can define the set   for a given integer   as the set of integers   for which there exist natural numbers   and   satisfying the Diophantine equation[1]

 , where  
 

An  -Kaprekar number for base   is then one which lies in the set  .

It was shown in 2000[1] that there is a bijection between the unitary divisors of   and the set   defined above. Let   denote the multiplicative inverse of   modulo  , namely the least positive integer   such that  , and for each unitary divisor   of   let   and  . Then the function   is a bijection from the set of unitary divisors of   onto the set  . In particular, a number   is in the set   if and only if   for some unitary divisor   of  .

The numbers in   occur in complementary pairs,   and  . If   is a unitary divisor of   then so is  , and if   then  .

Kaprekar numbers for edit

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

Let   and   be natural numbers, the number base  , and  . Then:

  •   is a Kaprekar number.
Proof

Let

 

Then,

 


The two numbers   and   are

 
 

and their sum is

 

Thus,   is a Kaprekar number.

  •   is a Kaprekar number for all natural numbers  .
Proof

Let

 

Then,

 

The two numbers   and   are

 
 

and their sum is

 

Thus,   is a Kaprekar number.

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

Let  ,  , and   be natural numbers, the number base  , and the power  . Then:

  •   is a Kaprekar number.
  •   is a Kaprekar number.

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

Let  ,  , and   be natural numbers, the number base  , and the power  . Then:

  •   is a Kaprekar number.
  •   is a Kaprekar number.

b = m2k + m2m + 1 and p = mn + 1 edit

Let  ,  , and   be natural numbers, the number base  , and the power  . Then:

  •   is a Kaprekar number.
  •   is a Kaprekar number.

b = m2k + m2m + 1 and p = mn + m − 1 edit

Let  ,  , and   be natural numbers, the number base  , and the power  . Then:

  •   is a Kaprekar number.
  •   is a Kaprekar number.

Kaprekar numbers and cycles of for specific , edit

All numbers are in base  .

Base   Power   Nontrivial Kaprekar numbers  ,   Cycles
2 1 10  
3 1 2, 10  
4 1 3, 10  
5 1 4, 5, 10  
6 1 5, 6, 10  
7 1 3, 4, 6, 10  
8 1 7, 10 2 → 4 → 2
9 1 8, 10  
10 1 9, 10  
11 1 5, 6, A, 10  
12 1 B, 10  
13 1 4, 9, C, 10  
14 1 D, 10  
15 1 7, 8, E, 10

2 → 4 → 2

9 → B → 9

16 1 6, A, F, 10  
2 2 11  
3 2 22, 100  
4 2 12, 22, 33, 100  
5 2 14, 31, 44, 100  
6 2 23, 33, 55, 100

15 → 24 → 15

41 → 50 → 41

7 2 22, 45, 66, 100  
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  
3 4 121, 2102, 2222, 10000  
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  
3 8 2012021, 10121020, 12101210, 21121001, 20210202, 22222222, 100000000  
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

Notes edit

  1. ^ a b Iannucci (2000)

References edit

  • 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.