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Pyridinium refers to the cation [C 5 H 5 NH] +. It is the conjugate acid of pyridine. Many related cations are known involving substituted pyridines, e.g. picolines ...
An ylide (/ ˈ ɪ l aɪ d /) [1] or ylid (/ ˈ ɪ l ɪ d /) is a neutral dipolar molecule containing a formally negatively charged atom (usually a carbanion) directly attached to a heteroatom with a formal positive charge (usually nitrogen, phosphorus or sulfur), and in which both atoms have full octets of electrons.
In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product: r a d ( n ) = ∏ p ∣ n p prime p {\displaystyle \displaystyle \mathrm {rad} (n)=\prod _{\scriptstyle p\mid n \atop p{\text{ prime}}}p}
These formulas may be applied repeatedly to get a simple expression for any cyclotomic polynomial () in terms of a cyclotomic polynomial of square free index: If q is the product of the prime divisors of n (its radical), then [5]
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
In number theory, quadratic integers are a generalization of the usual integers to quadratic fields.A complex number is called a quadratic integer if it is a root of some monic polynomial (a polynomial whose leading coefficient is 1) of degree two whose coefficients are integers, i.e. quadratic integers are algebraic integers of degree two.
Moreover, if the polynomial degree is a power of 2 and the roots are all real, then if there is a root that can be expressed in real radicals it can be expressed in terms of square roots and no higher-degree roots, as can the other roots, and so the roots are classically constructible. Casus irreducibilis for quintic polynomials is discussed by ...
The resonance structures below show the 1,3-dipole contribution, in which the two carbon atoms adjacent to the nitrogen have a negative or positive charge. [1] The most common representation of azomethine ylides is that in which the nitrogen is positively charged, and the negative charge is shared between the two carbon atoms.