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.

Kaprekar transformation, Kaprekar loops, Kaprekar constants.