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  2. Cyclic number - Wikipedia

    en.wikipedia.org/wiki/Cyclic_number

    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.

  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. Full reptend prime - Wikipedia

    en.wikipedia.org/wiki/Full_reptend_prime

    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.

  5. Reciprocals of primes - Wikipedia

    en.wikipedia.org/wiki/Reciprocals_of_primes

    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.

  6. Cyclic group - Wikipedia

    en.wikipedia.org/wiki/Cyclic_group

    A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.

  7. Midy's theorem - Wikipedia

    en.wikipedia.org/wiki/Midy's_theorem

    Also b n −1 is not a multiple of p for any value of n less than ℓ, because otherwise the repeating period of a/p in base b would be less than ℓ. Now suppose that ℓ = hk. Then b ℓ − 1 is a multiple of b k − 1. (To see this, substitute x for b k; then b ℓ = x h and x − 1 is a factor of x h − 1.

  8. Fermat quotient - Wikipedia

    en.wikipedia.org/wiki/Fermat_quotient

    If the base a is coprime to the exponent p then Fermat's little theorem says that q p (a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then q p (a) will be a cyclic number, and p will be a full reptend prime.

  9. Conjugacy class - Wikipedia

    en.wikipedia.org/wiki/Conjugacy_class

    A cyclic permutation of three (other one remains unchanged). Cycle type = [1 1 3 1]. Order = 3. ... is equal to the number of integer partitions of . This is ...