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Keywords = Kaprekar constants

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31 pages, 330 KiB  
Article
On Some Formulas for Kaprekar Constants
by Atsushi Yamagami and Yūki Matsui
Symmetry 2019, 11(7), 885; https://doi.org/10.3390/sym11070885 - 5 Jul 2019
Cited by 2 | Viewed by 3981
Abstract
Let b 2 and n 2 be integers. For a b-adic n-digit integer x, let A (resp. B) be the b-adic n-digit integer obtained by rearranging the numbers of all digits of x in descending [...] Read more.
Let b 2 and n 2 be integers. For a b-adic n-digit integer x, let A (resp. B) be the b-adic n-digit integer obtained by rearranging the numbers of all digits of x in descending (resp. ascending) order. Then, we define the Kaprekar transformation T ( b , n ) ( x ) : = A B . If T ( b , n ) ( x ) = x , then x is called a b-adic n-digit Kaprekar constant. Moreover, we say that a b-adic n-digit Kaprekar constant x is regular when the numbers of all digits of x are distinct. In this article, we obtain some formulas for regular and non-regular Kaprekar constants, respectively. As an application of these formulas, we then see that for any integer b 2 , the number of b-adic odd-digit regular Kaprekar constants is greater than or equal to the number of all non-trivial divisors of b. Kaprekar constants have the symmetric property that they are fixed points for recursive number theoretical functions T ( b , n ) . Full article
(This article belongs to the Special Issue Number Theory and Symmetry)
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