In mathematics, a Kaprekar number is a number that, in a given base, when squared, can be split into two numbers with the same number of digits as the original number which add up to the original number again. For example, the 3-digit number 703 is a Kaprekar number, because 703² = 494209, which can be split into 494 and 209, and 494 + 209 = 703. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...

Stated mathematically, let X be an integer with m digits when written in base n. If there exists two integers A and B satisfying the following conditions, then X is a Kaprekar number:

X² = An^m + B

X = A + B

The first few Kaprekar numbers in base 10 are (sequence A006886 in OEIS): The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

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 numbers are Kaprekar numbers. Jump to: navigation, search Look up one on Wiktionary, the free dictionary. ... Jump to: navigation, search 9 (nine) is the natural number following 8 and preceding 10. ... 45 is the natural number following 44 and followed by 46. ... 55 is the natural number following 54 and preceding 56. ... Jump to: navigation, search 99 (ninety-nine) is the natural number following 98 and preceding 100. ... The number 142,857 has some interesting mathematical properties. ... Jump to: navigation, search The binary numeral system represents numeric values using two symbols, typically 0 and 1. ... Jump to: navigation, search In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, excluding itself. ...

The Kaprekar numbers are named after D. R. Kaprekar. Shri Dattathreya Ramachandra Kaprekar (1905- 1986) was an Indian mathematician, whose name is associated with a number of concepts in number theory. ...

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