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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 …
Therefore, the base b expansion of / repeats the digits of the corresponding cyclic number infinitely, as does that of / with rotation of the digits for any a between 1 and p − 1. The cyclic number corresponding to prime p will possess p − 1 digits if and only if p is a full reptend prime.
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.
The multiple is thus the repeating digits of the fraction 3 ⁄ F, say b***a. In order for this cyclic permutation to take place, then 3 shall be the next remainder in the long division for 1 ⁄ F. Thus F shall be 7 as 1 × 10 ÷ 7 gives remainder 3. This yields the results that: X = the repeating digits of 1 ⁄ 7 =142857, and
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 = 0. 142857.
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.
Given a number base , a natural number with digits is an automorphic number if is a fixed point of the polynomial function = over /, the ring of integers modulo.As the inverse limit of / is , the ring of -adic integers, automorphic numbers are used to find the numerical representations of the fixed points of () = over .