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where b is the number base (10 for decimal), and p is a prime that does not divide b. (Primes p that give cyclic numbers in base b are called full reptend primes or long primes in base b). For example, the case b = 10, p = 7 gives the cyclic number 142857, and the case b = 12, p = 5 gives the cyclic number 2497.
Therefore, the base b expansion of / repeats the digits of the corresponding cyclic number infinitely, as does that of / with rotation of the digits for any a between 1 and p − 1. The cyclic number corresponding to prime p will possess p − 1 digits if and only if p is a full reptend prime.
A cyclic number [1] [2] is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic. [3] Any prime number is clearly cyclic. All cyclic numbers are square-free. [4] Let n = p 1 p 2 …
where d is the first digit of N and m is the number of digits. This explains the above common gcd and the phenomenon is true in any base if 10 is replaced by b, the base. The cyclic permutations are thus related to repeating decimals, the corresponding fractions, and divisors of 10 m −1. For examples the related fractions to the above cyclic ...
A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 10). ... 1, for some prime number n. 3 ... digit d. 2n+1: 3 ...
The Natural Area Code, this is the smallest base such that all of 1 / 2 to 1 / 6 terminate, a number n is a regular number if and only if 1 / n terminates in base 30. 32: Duotrigesimal: Found in the Ngiti language. 33: Use of letters (except I, O, Q) with digits in vehicle registration plates of Hong Kong. 34
If p is a proper prime ending in a 1, that is, if the repetend of 1 / p is a cyclic number of length p − 1 and p = 10h + 1 for some h, then each digit 0, 1, ..., 9 appears in the repetend exactly h = p − 1 / 10 times. For some other properties of repetends, see also. [15]
[1] [2] For example, 1193 is a circular prime, since 1931, 9311 and 3119 all are also prime. [3] A circular prime with at least two digits can only consist of combinations of the digits 1, 3, 7 or 9, because having 0, 2, 4, 6 or 8 as the last digit makes the number divisible by 2, and having 0 or 5 as the last digit makes it divisible by 5. [4]