On 2-adic Kaprekar constants and 2-digit Kaprekar distances
| Autor: | Atsushi Yamagami |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Discrete mathematics
Algebra and Number Theory Mathematics::Number Theory 010102 general mathematics Mersenne prime Prime number 02 engineering and technology 01 natural sciences Loop (topology) Integer 0202 electrical engineering electronic engineering information engineering Additive number theory Order (group theory) 020201 artificial intelligence & image processing 0101 mathematics Constant (mathematics) Mathematics Kaprekar number |
| Zdroj: | Journal of Number Theory. 185:257-280 |
| ISSN: | 0022-314X |
| DOI: | 10.1016/j.jnt.2017.09.004 |
| Popis: | 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 transformation T ( 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 ( c 0 ) b for all b ≥ 2 and 1 ≤ c ≤ b − 1 . |
| Databáze: | OpenAIRE |
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