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Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number. For example: 1/7 = 0.142857 142857...
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 …
A circular prime is a prime number with the property that the number generated at each intermediate step when cyclically permuting its (base 10) digits will be prime. [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 ...
If p is a prime number, then any group with p elements is isomorphic to the simple group Z/pZ. A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when gcd(n, φ(n)) = 1. [13] The sequence of cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd ...
A prime number (or prime) is a ... Some sources only list the smallest prime in each cycle, for example, ... gives a cyclic number. They are also called full reptend ...
The trivial group is the only group of order one, and the cyclic group C p is the only group of order p. There are exactly two groups of order p 2, both abelian, namely C p 2 and C p × C p. For example, the cyclic group C 4 and the Klein four-group V 4 which is C 2 × C 2 are both 2-groups of order 4.
For each odd prime power the corresponding factor (/) is the cyclic group of order () =, which may further factor into cyclic groups of prime-power orders. For powers of 2 the factor ( Z / 2 k Z ) × {\displaystyle (\mathbb {Z} /{2^{k}}\mathbb {Z} )^{\times }} is not cyclic unless k = 0, 1, 2, but factors into cyclic groups as described above.
Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order pq is cyclic when q < p are primes with p − 1 not divisible by q. For a necessary and sufficient condition, see cyclic number.