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Stated mathematically, let ''X'' be an 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''
: ''X''² = ''Abn'' + ''B''
: ''X'' = ''A'' + ''B''

The first few Kaprekar numbers in base 10 are :

: 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 Perfect Number s 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