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Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number. For example: 1/7 = 0.142857 142857...
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:
Because of the tremendous diversity allowed, in combination, by the valences of common atoms and their ability to form rings, the number of possible cyclic structures, even of small size (e.g., < 17 total atoms) numbers in the many billions. Cyclic compound examples: All-carbon (carbocyclic) and more complex natural cyclic compounds
A lactam is a cyclic amide, and beta-lactams are named so because the nitrogen atom is attached to the β-carbon atom relative to the carbonyl. The simplest β-lactam possible is 2-azetidinone. β-lactams are significant structural units of medicines as manifested in many β-lactam antibiotics . [ 2 ]
The bonding in carbon dioxide (CO 2): all atoms are surrounded by 8 electrons, fulfilling the octet rule.. The octet rule is a chemical rule of thumb that reflects the theory that main-group elements tend to bond in such a way that each atom has eight electrons in its valence shell, giving it the same electronic configuration as a noble gas.
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Since a linear order induces a cyclic order, cyclically ordered groups are also a generalization of linearly ordered groups: the rational numbers Q, the real numbers R, and so on. Some of the most important cyclically ordered groups fall into neither previous category: the circle group T and its subgroups , such as the subgroup of rational points .