In mathematics, a natural number in a given number base is a p {\displaystyle p} -Kaprekar number if the representation of its square in that base can be split into two partitions with the 2nd partition being p {\displaystyle p} digits in length, 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 {\displaystyle n} be a natural number. Then the Kaprekar function for base b > 1 {\displaystyle b>1} and power p > 0 {\displaystyle p>0} F p , b : N → N {\displaystyle F_{p,b}:\mathbb {N} \rightarrow \mathbb {N} } is defined to be the following:
F p , b ( n ) = α + β {\displaystyle F_{p,b}(n)=\alpha +\beta } , where β = n 2 mod b p {\displaystyle \beta =n^{2}{\bmod {b}}^{p}} and
α = n 2 − β b p {\displaystyle \alpha ={\frac {n^{2}-\beta }{b^{p}}}}
It can also be expressed as:
F p , b ( n ) = ⌊ n 2 b p ⌋ ( 1 − b p ) + n 2 {\displaystyle F_{p,b}(n)=\left\lfloor {\frac {n^{2}}{b^{p}}}\right\rfloor (1-b^{p})+n^{2}}
A natural number n {\displaystyle n} is a p {\displaystyle p} -Kaprekar number if it is a fixed point for F p , b {\displaystyle F_{p,b}} , which occurs if F p , b ( n ) = n {\displaystyle F_{p,b}(n)=n} . 0 {\displaystyle 0} and 1 {\displaystyle 1} are trivial Kaprekar numbers for all b {\displaystyle b} and p {\displaystyle p} , all other Kaprekar numbers are nontrivial Kaprekar numbers. The earlier example of 45 satisfies this definition with b = 10 {\displaystyle b=10} and p = 2 {\displaystyle p=2} , because
β = n 2 mod b p = 45 2 mod 1 0 2 = 25 {\displaystyle \beta =n^{2}{\bmod {b}}^{p}=45^{2}{\bmod {1}}0^{2}=25}
α = n 2 − β b p = 45 2 − 25 10 2 = 20 {\displaystyle \alpha ={\frac {n^{2}-\beta }{b^{p}}}={\frac {45^{2}-25}{10^{2}}}=20}
F 2 , 10 ( 45 ) = α + β = 20 + 25 = 45 {\displaystyle F_{2,10}(45)=\alpha +\beta =20+25=45}
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