On 2-adic Kaprekar constants and 2-digit Kaprekar distances

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 number obtained by rearranging the numbers of all digits of x in descending (resp. ascending) order. Then we define the Kaprekar transformationT(b,n)(x):=A−B. Then there exist the smallest integers d≥0 and ℓ≥1 such that T(b,n)d(x)↦⋯↦T(b,n)d+ℓ(x)=T(b,n)d(x). This loop is called the Kaprekar loop arising from x of length ℓ and the integer d is called the Kaprekar distance from x to the loop. In particular, if T(b,n)(x)=x, then x is called Kaprekar constant. In this article, we prove that any 2-adic Kaprekar constant is the 2-adic expression of a product of two suitable Mersenne numbers. As a corollary to this theorem, we see that for a prime number p, the b-adic expression of p is a b-adic Kaprekar constant if and only if b=2 and p is a Mersenne prime number. We also obtain some formulas for the Kaprekar distances from b-adic 2-digit integers of the form (c0)b for all b≥2 and 1≤c≤b−1.