On Some Formulas for Kaprekar Constants

• Main Authors: Atsushi Yamagami, Yūki Matsui
• Format: Article
• Language: English
• Published: MDPI AG 2019-07-01
• Series: Symmetry
• Subjects: Kaprekar constants; Kaprekar transformation; fixed points for recursive functions
• Online Access: https://www.mdpi.com/2073-8994/11/7/885

Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mo>&#8805;</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>n</mi> <mo>&#8805;</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> be integers. For a <i>b</i>-adic <i>n</i>-digit integer <i>x</i>, let <i>A</i> (resp. <i>B</i>) be the <i>b</i>-adic <i>n</i>-digit integer obtained by rearranging the numbers of all digits of <i>x</i> in descending (resp. ascending) order. Then, we define the <i>Kaprekar transformation</i> <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mi>A</mi> <mo>&#8722;</mo> <mi>B</mi> </mrow> </semantics> </math> </inline-formula>. If <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> </mrow> </semantics> </math> </inline-formula>, then <i>x</i> is called a <i>b</i>-<i>adic</i> <i>n</i>-<i>digit Kaprekar constant</i>. Moreover, we say that a <i>b</i>-adic <i>n</i>-digit Kaprekar constant <i>x</i> is <i>regular</i> when the numbers of all digits of <i>x</i> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>b</mi> <mo>&#8805;</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, the number of <i>b</i>-adic odd-digit regular Kaprekar constants is greater than or equal to the number of all non-trivial divisors of <i>b</i>. Kaprekar constants have the symmetric property that they are fixed points for recursive number theoretical functions <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mrow> <mo>(</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </msub> </semantics> </math> </inline-formula>.