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  2. Cyclic number - Wikipedia

    en.wikipedia.org/wiki/Cyclic_number

    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 ...

  3. Cyclic number (group theory) - Wikipedia

    en.wikipedia.org/wiki/Cyclic_number_(group_theory)

    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 …

  4. Cyclic group - Wikipedia

    en.wikipedia.org/wiki/Cyclic_group

    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:

  5. Cyclic (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Cyclic_(mathematics)

    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 ...

  6. 142857 - Wikipedia

    en.wikipedia.org/wiki/142857

    142857 × 7 4 = 342999657 342 + 999657 = 999999. If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number. [citation needed] 857 2 = 734449 142 2 = 20164 734449 − 20164 = 714285. It is the repeating part in the decimal expansion of the rational number ⁠ 1 / 7 ...

  7. Cyclic algebra - Wikipedia

    en.wikipedia.org/wiki/Cyclic_algebra

    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 ]

  8. Cycles and fixed points - Wikipedia

    en.wikipedia.org/wiki/Cycles_and_fixed_points

    GB has 2 fixed points and 2 7-cycles P * (1,2,3,4) T = (4,1,3,2) T Permutation of four elements with 1 fixed point and 1 3-cycle In mathematics , the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S .

  9. Cycle graph (algebra) - Wikipedia

    en.wikipedia.org/wiki/Cycle_graph_(algebra)

    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 ...

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