Consider an n-digit number k. Square it, and then add the right n digits to the left n or n-1 digits (by convention, the second part may start with the digit 0, but must be nonzero). If the result is k then it is called a Kaprekar number. They are named after D. R. Kaprekar, a recreational mathematician from India.
We can extend the definition to any base b:
Let X be a non-negative integer and n a positive integer. X is an n-Kaprekar number for base b if there exist non-negative integer A, and positive integer B satisfying:
- X2 = Abn + B, where 0 < B < bn
- X = A + B
Examples in Base 10
297: 2972 = 88209, which can be split into 88 and 209, and 88 + 209 = 297.
- 999: 9992 = 998001, which can be split into 998 and 001, and 998 + 001 = 999.
- In particular, 9, 99, 999… are all Kaprekar numbers.
- More generally, for any base b, there exist infinitely many Kaprekar numbers, including all numbers of the form bn − 1.
100: 100 is NOT a Kaprekar number as, although 1002 = 10000 and 100 + 00 = 100, the second part here is zero.
(P.S. Another post on Kaprekar is coming soon!)