Kaprekar number

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

The earlier example of 45 satisfies this definition with ${\displaystyle b=10}$ and ${\displaystyle p=2}$, 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 ${\displaystyle m>0}$ then

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

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 ${\displaystyle n\leq b^{p}}$ 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

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 the Diophantine equation

${\displaystyle X^{2}=AN B}$, where ${\displaystyle 0\leq B$

${\displaystyle X=A B}$

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 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 ${\displaystyle X=d\operatorname {Inv} (d,e)}$ then ${\displaystyle N-X=e\operatorname {Inv} (e,d)}$.