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 example, 297 is a Kaprekar number for base 10, because 297² = 88209, which can be split into 88 and 209, and 88 + 209 = 297. The second part may start with the digit 0, but must be positive. 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 not positive.

Stated mathematically, let X be a non-negative integer. X is a Kaprekar number for base b if there exist non-negative integers n, A and B satisfying the following three conditions:

0 < B < bⁿ

X² = Abⁿ + B

X = A + B

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

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

In binary, all even perfect numbers are Kaprekar numbers.

For any base there exist infinitely many Kaprekar numbers; in particular, for base b all numbers of the form bn - 1 are Kaprekar numbers.

The Kaprekar numbers are named after D. R. Kaprekar.

References

* D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.

* M. Charosh, Some Applications of Casting Out 999...'s, Journal of Recreational Mathematics 14, 1981-82, pp. 111-118

* Douglas E. Iannucci, The Kaprekar Numbers, Journal of Integer Sequences, Vol. 3 (2000), http://www.math.uwaterloo.ca/JIS/VOL3/iann2a.html