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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.
Every cycle graph is a circulant graph, as is every crown graph with number of vertices congruent to 2 modulo 4.. The Paley graphs of order n (where n is a prime number congruent to 1 modulo 4) is a graph in which the vertices are the numbers from 0 to n − 1 and two vertices are adjacent if their difference is a quadratic residue modulo n.
[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]
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 …
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 cycle graph is: 2-edge colorable, if and only if it has an even number of vertices; 2-regular; 2-vertex colorable, if and only if it has an even number of vertices. More generally, a graph is bipartite if and only if it has no odd cycles (Kőnig, 1936). Connected; Eulerian; Hamiltonian; A unit distance graph; In addition:
Cyclic module, a module generated by a single element; Cyclic notation, a way of writing permutations; 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 ...
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.