JP Journal of Algebra, Number Theory and Applications
Volume 40, Issue 6, Pages 957 - 1028
(December 2018) http://dx.doi.org/10.17654/NT040060957 |
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ON 3-ADIC KAPREKAR LOOPS
Atsushi Yamagami and YÅ«ki Matsui
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Abstract: Let b ≥ 2 and n ≥ 2 be any 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 xin descending (resp. ascending) order. We define the Kaprekar transformation T(b, n)(x) : = A – B. Then there exist the smallest integers d(x) ≥ 0 and such that This loop is called the Kaprekar loop arising from x. In this article, we reveal the structure of the 3-adic Kaprekar loops by obtaining the formulas for the number of all 3-adic n-digit Kaprekar loops and their lengths in terms of n. |
Keywords and phrases: Kaprekar transformation, Kaprekar loops, Kaprekar constants.
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