Kaprekar number

Kaprekar number

Named after Dattaraya Ramchandra Kaprekar .

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.

Examples:

297 is a Kaprekar number for base 10, because 297² = 88209, which can be split into 88 and 209, and 88 + 209 = 297.

45 is a Kaprekar number, because 45² = 2025 and 20+25 = 45.

Let X be a non-negative integer.

X is a Kaprekar number for base b if there exist non-negative integers n, A, and positive number B satisfying:

X² = Abn + B, where 0 < B < bn

X = A + B

Note that X is also a Kaprekar number for base bn, for this specific choice of n. More narrowly, we can define the set K(N) for a given integer N as the set of integers X for which[1]

X² = AN + B, where 0 < B < N

X = A + B

Each Kaprekar number X for base b is then counted in one of the sets K(b), K(b²), K(b³),….

Note:

By convention, the second part may start with the digit 0, but must be nonzero.

For example, 999 is a Kaprekar number for base 10, because 999² = 998001, which can be split into 998 and 001, and 998 + 001 = 999. But 100 is not; although 100² = 10000 and 100 + 00 = 100, the second part here is zero.

Reference: https://en.wikipedia.org/wiki/Kaprekar_number