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Group (mathematics) The manipulations of the Rubik's Cube form the Rubik's Cube group. In mathematics, a group is a set with an operation that satisfies the following constraints: the operation is associative and has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with ...
The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q(ω) – the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial. In particular, ω satisfies the equation.
In the 1950s, Hillel Furstenberg introduced a proof by contradiction using point-set topology. Define a topology on the integers Z, called the evenly spaced integer topology, by declaring a subset U ⊆ Z to be an open set if and only if it is either the empty set, ∅, or it is a union of arithmetic sequences S(a, b) (for a ≠ 0), where
Definition. An abundant number is a natural number n for which the sum of divisors σ(n) satisfies σ(n) > 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) satisfies s(n) > n.
The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain. Most of the identities following below are provided by Simon Plouffe. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high ...
Irrational number. The number √ 2 is irrational. In mathematics, the irrational numbers ( in- + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also ...
In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n .
1 Control-C has typically been used as a "break" or "interrupt" key. 2 Control-D has been used to signal "end of file" for text typed in at the terminal on Unix / Linux systems. Windows, DOS, and older minicomputers used Control-Z for this purpose. 3 Control-G is an artifact of the days when teletypes were in use.