In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a 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 – 0288 = 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: The sequence is called a Kaprekar sequence and the function 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, 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, 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. Note that the numbers \alpha and \beta have the same digit sum and hence the same remainder modulo b - 1. Therefore, each number in a Kaprekar sequence of base b numbers (other than possibly the first) is a multiple of b - 1. When leading zeroes are retained, only repdigits lead to the trivial Kaprekar's constant.

Families of Kaprekar's constants

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.

b = 2k

It can be shown that all natural numbers are fixed points of the Kaprekar mapping in even base b = 2k for all natural numbers n.

Citations