Kaprekar number

In mathematics, a Kaprekar number (By: Shri Dattatreya Ramachandra Kaprekar, 1905–1986, Indian mathematician) is that non-negative integer such that, in a given base, the digits of its square on that basis they can be separated into two numbers which added together give the original number. The simplest example is 9, its square is 81 and 8+1= 9. Another example is the number 703, its square is 494209. If we separate 494209 into two new numbers, 494 and 209, we get that 494 + 209 = 703. Similarly, the number 297 is also a Kaprekar number, since it is possible decompose the square 297² = 88209 into 88 and 209.
The second number can start with zero, but it must be positive. An example is 999, since 999²=998001 and decomposes into 998 and 001. For this reason, the number 100 is not a Kaprekar number, since 100²=10000 and decomposes into 100 + 00, but the second addend is not positive.

Mathematically, let X be a non-negative integer. X is a Kaprekar number for base b if there are n non-negative integers, A and B, which satisfy the following conditions:

0 ” B. bⁿ

X2 = Abⁿ + B

X = A + B

The first Kaprekar numbers in base 10 are ((sequence A006886 in OEIS)):

1, 9, 45, 55, 99, 297, 703,...

In binary (base 2) all perfect numbers are Kaprekar numbers.

In any base there are infinitely many Kaprekar numbers, in particular, given a base b, all numbers of the form bⁿ-1 are Kaprekar numbers.

Kaprekar numbers are named in honor of D. R. Kaprekar. They should not be confused with the Kaprekar constant, which is the number 6174.