Base-dependent property of integers
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
Let
be a natural number. Then the Kaprekar function for base
and power
is defined to be the following:
,
where
and
![{\displaystyle \alpha ={\frac {n^{2}-\beta }{b^{p}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8441e0a6e512b97b37668b58ccffd1a9f4506e17)
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
![{\displaystyle \beta =n^{2}{\bmod {b}}^{p}=45^{2}{\bmod {1}}0^{2}=25}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0288e6507a0bd7ed9de38c972c03d837fa78e5d9)
![{\displaystyle \alpha ={\frac {n^{2}-\beta }{b^{p}}}={\frac {45^{2}-25}{10^{2}}}=20}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11583e478d5395cac8d5fe875abf3827e8acc566)
![{\displaystyle F_{2,10}(45)=\alpha +\beta =20+25=45}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e787803f676943a071c2f5207e9240e60ec101de)
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
![{\displaystyle {\begin{aligned}n^{2}&=(b^{p}+m)^{2}\\&=b^{2p}+2mb^{p}+m^{2}\\&=(b^{p}+2m)b^{p}+m^{2}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b57eab02f80af766bdb35a3996e8cf732087a344)
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
The set
for a given integer
can be defined as the set of integers
for which there exist natural numbers
and
satisfying the Diophantine equation[1]
, where ![{\displaystyle 0\leq B<N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f3c074a1a335f2bf849e789018473e46be4822)
![{\displaystyle X=A+B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af318676df0ef92628ca7407b1ea1e46854eb80e)
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
b = 4k + 3 and p = 2n + 1
Let
and
be natural numbers, the number base
, and
. Then:
is a Kaprekar number.
Proof Let
Then,
The two numbers
and
are
![{\displaystyle \beta =X_{1}^{2}{\bmod {b}}^{p}=k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0eaa82c5685878e3af839aa801ed499bdb05aab)
![{\displaystyle \alpha ={\frac {X_{1}^{2}-\beta }{b^{p}}}=k+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/398ba80eee26fcfd9bfdb9cdd720b742cd829af5)
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
![{\displaystyle \beta =X_{2}^{2}{\bmod {b}}^{p}=k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cf9fa962a3e23800d463effd4ccf3bec2db3aed)
![{\displaystyle \alpha ={\frac {X_{2}^{2}-\beta }{b^{p}}}=k+1+\sum _{i=1}^{n}(kb+(3k+2))b^{2i-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c33c89942e7e5a8c80695ad8bf6dd3af94794d)
and their sum is
Thus,
is a Kaprekar number.
b = m2k + m + 1 and p = mn + 1
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
Let
,
, and
be natural numbers, the number base
, and the power
. Then:
is a Kaprekar number.
is a Kaprekar number.
b = m2k + m2 − m + 1 and p = mn + 1
Let
,
, and
be natural numbers, the number base
, and the power
. Then:
is a Kaprekar number.
is a Kaprekar number.
b = m2k + m2 − m + 1 and p = mn + m − 1
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
,
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
Kaprekar numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
See also
Notes
References
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Possessing a specific set of other numbers |
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Expressible via specific sums |
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