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| {{distinguish|Kaprekar's constant}}
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| In [[mathematics]], a '''Kaprekar number''' for a given [[base (exponentiation)|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]].
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| == Definition ==
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| 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:
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| : ''X''² = ''Ab<sup>n</sup>'' + ''B'', where 0 < ''B'' < ''b<sup>n</sup>''
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| : ''X'' = ''A'' + ''B''
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| Note that ''X'' is also a Kaprekar number for base ''b<sup>n</sup>'', 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<ref name=iannucci>Ianucci (2000)</ref>
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| : ''X''² = ''AN'' + ''B'', where 0 < ''B'' < ''N''
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| : ''X'' = ''A'' + ''B''
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| Each Kaprekar number ''X'' for base ''b'' is then counted in one of the sets ''K(b)'', ''K(b²)'', ''K(b³)'',….
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| == Examples ==
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| 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 [[Positive number|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.
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| The first few Kaprekar numbers in base 10 are:
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| :[[1 (number)|1]], [[9 (number)|9]], [[45 (number)|45]], [[55 (number)|55]], [[99 (number)|99]], 297, 703, [[999 (number)|999]] , 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, [[9999 (number)|9999]] , 17344, 22222, 38962, 77778, 82656, 95121, 99999, [[142857 (number)|142857]], 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, ... {{OEIS|id=A006886}} | |
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| 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 ''b<sup>n</sup>'' - 1.
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| == Properties ==
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| * It was shown in 2000<ref name=iannucci/> that the Kaprekar numbers for base ''b'' are in bijection with the [[unitary divisor]]s of ''b<sup>n</sup>'' − 1, in the following sense. Let Inv(''a,b'') denote the [[Modular multiplicative inverse|multiplicative inverse]] of ''a'' modulo ''b'', namely the least positive integer ''m'' such that <math>am \equiv 1 \pmod b</math>. Then, a number ''X'' is in the set ''K(N)'' (defined above) if and only if ''X'' = ''d'' Inv(''d, (N-1)/d'') for some unitary divisor ''d'' of ''N''-1. In particular,
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| ** For each ''X'' in ''K(N)'', ''N - X'' is in ''K(N)''.
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| ** In [[binary numeral system|binary]], all even [[perfect number]]s are Kaprekar numbers.
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| ==See also==
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| *[[Kaprekar]]
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| *[[Kaprekar's constant]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{cite journal |author=D. R. Kaprekar |title=On Kaprekar numbers |journal=[[Journal of Recreational Mathematics]] |volume=13 |year=1980-1981 |pages=81–82}}
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| * {{cite journal |author=M. Charosh |title=Some Applications of Casting Out 999...'s |journal=[[Journal of Recreational Mathematics]] |volume=14 |year=1981-1982 |pages=111–118}}
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| * {{cite journal |author=Douglas E. Iannucci |url=http://www.math.uwaterloo.ca/JIS/VOL3/iann2a.html |title=The Kaprekar Numbers |journal=[[Journal of Integer Sequences]] |volume=3 |year=2000 |pages=00.1.2}}
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| * [[Yutaka Nishiyama]] (2006). [http://plus.maths.org/issue38/features/nishiyama/index.html "Mysterious Number 6174"].
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| {{Classes of natural numbers}}
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| [[Category:Base-dependent integer sequences]]
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