Kaprekar number
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 instance, 45 is a Kaprekar number, because 452 = 2025 and 20 + 25 = 45. The Kaprekar numbers are named after D. R. Kaprekar.
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:
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
Each Kaprekar number X for base b is then counted in one of the sets K(b), K(b2), K(b3),….
297 is a Kaprekar number for base 10, because 2972 = 88209, which can be split into 88 and 209, and 88 + 209 = 297. 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.
The first few Kaprekar numbers in base 10 are:
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.
In base 12, the Kaprekar numbers are
In base 16, the Kaprekar numbers are: 1, 6, A, F, 33, 55, 5B, 78, 88, AB, CD, FF, 15F, 334, 38E, 492, 4ED, 7E0, 820, B13, B6E, C72, CCC, EA1, FA5, FFF, 191A, 2A2B, 3C3C, 4444, 5556, 6667, 7F80, 8080, 9999, AAAA, BBBC, C3C4, D5D5, E6E6, FFFF, 1745E, 20EC2, 2ACAB, 2D02E, 30684, 3831F, 3E0F8, 42108, 47AE1, 55555, 62FCA, 689A3, 7278C, 76417, 7A427, 7FE00, 80200, 85BD9, 89AE5, 89BE9, 8D874, 9765D, 9D036, AAAAB, AF0B0, B851F, BDEF8, C1F08, C7CE1, CF97C, D5355,...
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