On 3-Adic Kaprekar Loops

Yamagami, Atsushi; Matsui, Yutaka

doi:10.17654/nt040060957

Cited by 2 publications

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“…Proof. We see immediately that Parts (1)- (6) and (8) are implied by the respective formulas obtained in Theorem 3(1), (2), (4) and Theorem 4 for the respective digits n, since these formulas give distinct n-digit regular Kaprekar constants for distinct positive integers m, and we see that:…”

Section: Some Formulas For All N-digit Regular Kaprekar Constants Witmentioning

confidence: 57%

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On Some Formulas for Kaprekar Constants

Yamagami

Matsui

2019

Symmetry

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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 ) .

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Section: Some Formulas For All N-digit Regular Kaprekar Constants Witmentioning

confidence: 57%

“…In the case where b = 3, the authors [8] showed that for any n ≥ 2, all three-adic n-digit non-trivial Kaprekar constants are of the form:…”

Section: Examplementioning

confidence: 99%

On Some Formulas for Kaprekar Constants

Yamagami

Matsui

2019

Symmetry

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Symmetries in Generalized Kaprekar’s Routine

Nuez¹

2021

Symmetry

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In this paper, algebraic relations were established that determined the invariance of a transformed number after several transformations. The restrictions that determine the group structure of these relationships were analyzed, as was the case of the Klein group. Parametric Kr functions associated with the existence of cycles were presented, as well as the role of the number of their links in the grouping of numbers in higher-order equivalence classes. For this, we developed a methodology based on binary equivalence relations and the complete parameterization of the Kaprekar routine using Ki functions of parametric transformation.

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On 3-Adic Kaprekar Loops

Cited by 2 publications

References 0 publications

On Some Formulas for Kaprekar Constants

On Some Formulas for Kaprekar Constants

Symmetries in Generalized Kaprekar’s Routine

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