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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.
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 …
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 ...
The cyclic number corresponding to prime p will possess p − 1 digits if and only if p is a full reptend prime. That is, the multiplicative order ord p b = p − 1, which is equivalent to b being a primitive root modulo p. The term "long prime" was used by John Conway and Richard Guy in their Book of Numbers.
A full reptend prime, full repetend prime, proper prime [7]: 166 or long prime in base b is an odd prime number p such that the Fermat quotient = (where p does not divide b) gives a cyclic number with p − 1 digits.
Two elements g and h of the free group F on a set Y are conjugate if and only if, when they are written as products of elements y and y −1 with y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the rewriting rules that allow one to remove or add adjacent y and y −1.
A heterocyclic compound is a cyclic compound that has atoms of at least two different elements as members of its ring(s). [5] Cyclic compounds that have both carbon and non-carbon atoms present are heterocyclic carbon compounds, and the name refers to inorganic cyclic compounds as well (e.g., siloxanes, which contain only silicon and oxygen in ...
Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order of x. If the order of x is n , then x n = x 0 = 1 , {\displaystyle x^{n}=x^{0}=1,} and the cyclic group generated by x consists of the n first powers of x (starting indifferently from the exponent 0 or 1 ).