In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a four-digit random number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers. As an example, starting with the number 8991 in base 10:
9981 – 1899 = 8082 8820 – 288 = 8532 8532 – 2358 = 6174 7641 – 1467 = 6174 6174, known as Kaprekar's constant, is a fixed point of this algorithm. Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations. The algorithm runs on any natural number in any given number base.
Definition and properties The algorithm is as follows:
Choose any four-digit natural number n {\displaystyle n} in a given number base b {\displaystyle b} . This is the first number of the sequence. Create a new number α {\displaystyle \alpha } by sorting the digits of n {\displaystyle n} in descending order, and another number β {\displaystyle \beta } by sorting the digits of n {\displaystyle n} in ascending order. These numbers may have leading zeros, which can be ignored. Subtract α − β {\displaystyle \alpha -\beta } to produce the next number of the sequence. Repeat step 2. The sequence is called a Kaprekar sequence and the function K b ( n ) = α − β {\displaystyle K_{b}(n)=\alpha -\beta } is the Kaprekar mapping. Some numbers map to themselves; these are the fixed points of the Kaprekar mapping, and are called Kaprekar's constants. Zero is a Kaprekar's constant for all bases b {\displaystyle b} , and so is called a trivial Kaprekar's constant. All other Kaprekar's constants are nontrivial Kaprekar's constants. For example, in base 10, starting with 3524,
K 10 ( 3524 ) = 5432 − 2345 = 3087 {\displaystyle K_{10}(3524)=5432-2345=3087}
K 10 ( 3087 ) = 8730 − 378 = 8352 {\displaystyle K_{10}(3087)=8730-378=8352}
K 10 ( 8352 ) = 8532 − 2358 = 6174 {\displaystyle K_{10}(8352)=8532-2358=6174}
K 10 ( 6174 ) = 7641 − 1467 = 6174 {\displaystyle K_{10}(6174)=7641-1467=6174}
with 6174 as a Kaprekar's constant. All Kaprekar sequences will either reach one of these fixed points or will result in a repeating cycle. Either way, the end result is reached in a fairly small number of steps (within seven iterations or steps). Note that the numbers α {\displaystyle \alpha } and β {\displaystyle \beta } have the same digit sum and hence the same remainder modulo b − 1 {\displaystyle b-1} . Therefore, each number in a Kaprekar sequence of base b {\displaystyle b} numbers (other than possibly the first) is a multiple of b − 1 {\displaystyle b-1} . When leading zeroes are retained, only repdigits lead to the trivial Kaprekar's constant. In base 4, it can easily be shown that all numbers of the form 3021, 310221, 31102221, 3...111...02...222...1 (where the length of the "1" sequence and the length of the "2" sequence are the same) are fixed points of the Kaprekar mapping. In base 10, it can easily be shown that all numbers of the form 6174, 631764, 63317664, 6...333...17...666...4 (where the length of the "3" sequence and the length of the "6" sequence are the same) are fixed points of the Kaprekar mapping.
Determination of Kaprekar numbers In the following, "Kaprekar's constant k" refers to a number that becomes a positive fixed point k as a result of Kaprekar's routine. In 1981, G. D. Prichett, et al. showed that the Kaprekar's constants in base 10 are limited to two numbers, 495 (3 digits) and 6174 (4 digits). They also classified the Kaprekar numbers into four types, but there was some overlap in the classification. In 2005, Y. Hirata calculated all fixed points up to 31 decimal digits and examined their distribution. In 2024, Haruo Iwasaki of the Ranzan Mathematics Study Group (headed by Kenichi Iyanaga) showed that in order for a natural number to be a Kaprekar number, it must belong to one of five mutually disjoint sets composed of combinations of the seven numbers 495, 6174, 36, 123456789, 27, 875421 and 09. Iwasaki also showed that this new classification using the five sets includes a corrected classification by Prichett, et al. As a result, if n is considered as a constant, then the number of decimal n-digit Kaprekar numbers is determined by two types of equations:
(1) n = 3 x ( x ≥ 1 ) , {\displaystyle n=3x\quad \quad (x\geq 1)\,,} ......... For the sequence of x 3-digit constants 495 (2) n = 4 + 2 x ( x ≥ 0 ) , {\displaystyle n=4+2x\quad (x\geq 0)\,,} ...... Sequence of 4-digit constant 6174 followed by x 2-digit constants 36 or by three types of Diophantine equations:
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