Kierroslukujen

Kierroslukujen, often translated as "rotational numbers" or "cyclic numbers" in English, is a concept in number theory that refers to numbers whose multiples are cyclic permutations of each other. A number is a kierrosluku if its multiples, when written out, show the same digits in the same order but shifted cyclically.

The most famous example of a kierrosluku is the number 142857. When multiplied by 1, it is 142857. When multiplied by 2, it is 285714. Multiplying by 3 gives 428571, by 4 gives 571428, by 5 gives 714285, and by 6 gives 857142. The sequence of digits 1, 4, 2, 8, 5, 7 is preserved and cyclically shifted in each multiple.

Kierroslukuja are closely related to repeating decimals. Specifically, a number $n$ is a kierrosluku if and only if $1/n$ has a repeating decimal expansion where the length of the repeating part is $n-1$. In the case of 142857, $1/7 = 0.142857142857...$, and the repeating block has a length of 6, which is $7-1$.

The general form of kierroslukuja can be derived from prime numbers $p$ such that $10$ is a primitive root modulo $p$. These primes are called full reptend primes. The kierrosluku generated by such a prime $p$ is given by the integer represented by the repeating digits of $1/p$. The length of the repeating part will be $p-1$. For example, the next full reptend prime after 7 is 17. $1/17 = 0.0588235294117647...$, and the kierrosluku is 0588235294117647.

The concept of kierroslukuja has applications in recreational mathematics and in understanding the properties of decimal expansions of fractions. They are a fascinating illustration of the patterns that can emerge from arithmetic operations.

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