Kaprekar number

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 positive number "B" satisfying the following three conditions:

: 0 < "B" < "bn": "X"² = "Abn" + "B": "X" = "A" + "B"

The first few Kaprekar numbers in base 10 are OEIS|id=A006886:

: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. They should not be mistaken for number 6174, known as Kaprekar's constant.

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


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