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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. Not all values of p will yield a cyclic number using this formula; for example, the case b = 10, p = 13 gives 076923076923, and the case b = 12, p = 19 gives 076B45076B45076B45. These failed cases will always contain a ...
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 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 except 2. The cyclic numbers are:
In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field and plays a key role in the theory of central simple algebras. Definition [ edit ]
Cyclic number, a number such that cyclic permutations of the digits are successive multiples of the number; Cyclic order, a ternary relation defining a way to arrange a set of objects in a circle; Cyclic permutation, a permutation with one nontrivial orbit; Cyclic polygon, a polygon which can be given a circumscribed circle; Cyclic shift, also ...
A cyclic group Z n is a group all of whose elements are powers of a particular element a where a n = a 0 = e, the identity. A typical realization of this group is as the complex n th roots of unity. Sending a to a primitive root of unity gives an isomorphism between the two. This can be done with any finite cyclic group.
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.
In a finite group, some non-zero power of a must be the group identity, which we denote either as e or 1; the lowest such power is the order of the element a, the number of distinct elements in the cycle that it generates. In a cycle graph, the cycle is represented as a polygon, with its vertices representing the group elements and its edges ...