Kaprekar numbers

In mathematics, a Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split into two parts that add up to the original number again.

For example,

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170

... are the Kaprekar numbers

Because

9^2 = 81 and 8 + 1 = 9
45^2 = 2025 and 20 + 25 = 45
55^2 = 3025 and 30 + 25 = 55
99^2 = 9801 and 98 + 01 = 99
........................................
........................................
703^2 = 494209 and 494 + 209 = 703

Theorem 1:

k is in K(N) if and only if k = d Inv(d, (N-1)/d) for some unitary divisor d of N - 1

Theorem 2:

Every even perfect number is a Kaprekar number in the binary base.

http://www.cs.uwaterloo.ca/journals/JIS/VOL3/iann2a.html

(2)

The Indian mathematician D.R.Kaprekar made the following discovery in 1949.

(1) Take a four-digit number with different digits (acbd with a < b < c < d)
You'll get 6174 in no more than 7 steps.

Investigate in other bases.