On Some Formulas for Kaprekar Constants

Atsushi Yamagami,Yūki Matsui

doi:10.3390/sym11070885

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 (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-adicn-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 ) .

Highlights

IntroductionLet Z be the set of all rational integers. In this article, the symbol [α] with any rational number α stands for the greatest integer that is less than or equal to α
Proofs of Theorems and Corollaries in the Introduction2.1 A Proof of Theorem 1 . . . . . . . . . . . . . . . . . .2.2 A Proof of Corollary 1 . . . . . . . . . . . . . . . . .2.3 A Proof of Theorem 2 . . . . . . . . . . . . . . . . . .2.4 A Proof of Corollary 2 . . . . . . . . . . . . . . . . .On n-Digit Regular Kaprekar Constants with Specified n
As a corollary of Theorem 1, we immediately obtain some results on the positivity of the numbers νreg (b, n) of all b-adic n-digit regular Kaprekar constants as in the following: Corollary 1

Summary

Introduction

Let Z be the set of all rational integers. In this article, the symbol [α] with any rational number α stands for the greatest integer that is less than or equal to α. Since there is no two-digit non-regular Kaprekar constant by definition, we see immediately that for any integer b ≥ 2, νnon-reg (b, 2) = 0 and ν(b, 2) = νreg (b, 2). The aim of this article is to see formulas for b-adic n-digit regular and non-regular Kaprekar constants and to study the properties of νreg (b, n) and νnon-reg (b, n) towards answers to the questions above. As a corollary of Theorem 1, we immediately obtain some results on the positivity of the numbers νreg (b, n) of all b-adic n-digit regular Kaprekar constants as in the following: Corollary 1. As a corollary of Theorem 2, we immediately obtain the following result on the positivity of the numbers νreg (b, n) of all b-adic n-digit non-regular Kaprekar constants: Corollary 2. Theory and Symmetry,” since Kaprekar constants have the symmetric property that they are fixed points for recursive number theoretical functions T(b,n)

A Proof of Theorem 1

A Proof of Corollary 1

A Proof of Theorem 2

A Proof of Corollary 2

Some Formulas for All n-Digit Regular Kaprekar Constants with Specified n