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Størmer Number
In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer n for which the greatest prime factor of n^2+1 is greater than or equal to 2n. They are named after Carl Størmer. Sequence The first few Størmer numbers are: Density John Todd proved that this sequence is neither finite nor cofinite. More precisely, the natural density of the Størmer numbers lies between 0.5324 and 0.905. It has been conjectured that their natural density is the natural logarithm of 2, approximately 0.693, but this remains unproven. Because the Størmer numbers have positive density, the Størmer numbers form a large set. Application The Størmer numbers arise in connection with the problem of representing the Gregory numbers (arctangents of rational numbers) G_=\arctan\frac as sums of Gregory numbers for integers (arctangents of unit fractions). The Gregory number G_ may be decomposed by repeatedly multiplying the Gaussian integer In number theory, a Gaussian inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Størmer
Fredrik Carl Mülertz Størmer (3 September 1874 – 13 August 1957) was a Norwegian mathematician and astrophysicist. In mathematics, he is known for his work in number theory, including the calculation of and Størmer's theorem on consecutive smooth numbers. In physics, he is known for studying the movement of charged particles in the magnetosphere and the formation of aurorae, and for his book on these subjects, ''From the Depths of Space to the Heart of the Atom''. He worked for many years as a professor of mathematics at the University of Oslo in Norway. A crater on the far side of the moon is named after him. Personal life and career Størmer was born on 3 September 1874 in Skien, the only child of a pharmacist Georg Ludvig Størmer (1842–1930) and Elisabeth Amalie Johanne Henriette Mülertz (1844–1916). His uncle was the entrepreneur and inventor Henrik Christian Fredrik Størmer. Størmer studied mathematics at the Royal Frederick University in Kristiania, Nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
20 (number)
20 (twenty; Roman numeral XX) is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score. In mathematics *20 is a pronic number. *20 is a tetrahedral number as 1, 4, 10, 20. *20 is the basis for vigesimal number systems. *20 is the third composite number to be the product of a squared prime and a prime, and also the second member of the (''2''2)''q'' family in this form. *20 is the smallest primitive abundant number. *An icosahedron has 20 faces. A dodecahedron has 20 vertices. *20 can be written as the sum of three Fibonacci numbers uniquely, i.e. 20 = 13 + 5 + 2. *20 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case. (e.g. the newspaper headline "Scores of Typhoon Survivors Flown to Manila")."CBS News"''Scores of Typhoon Survivors Flown to Manila'' (November 2013) In sports * Twenty20 is a form of limited overs cricket where each team plays only 20 overs. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /math> or \Z Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic. Gaussian integers are algebraic integers and form the simplest ring of quadratic integers. Gaussian integers are named after the German mathematician Carl Friedrich Gauss. Basic definitions The Gaussian integers are the set :\mathbf \, \qquad \text i^2 = -1. In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers. Since the Gaussian integers are closed under addition and multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Fraction
A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/''n''. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. Arithmetic Elementary arithmetic Multiplying any two unit fractions results in a product that is another unit fraction: \frac1x \times \frac1y = \frac1. However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction: \frac1x + \frac1y = \frac \frac1x - \frac1y = \frac \frac1x \div \frac1y = \frac. Modular arithmetic In modular arithmetic, unit fractions can often be converted into equivalent integers using a calculation based on greatest common divisors. In turn, this conversion can be used to simplify division operations in modular arithmetic, by transforming them into equivalent multiplication operations. Specifically, consider the problem of dividing by a value x modulo y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arctangent
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. Notation Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of ''θ'' radians will correspond to an arc whose length is ''rθ'', where ''r'' is the radius of the circle. Thus in the unit cir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregory Number
In mathematics, a Gregory number, named after James Gregory, is a real number of the form: : G_x = \sum_^\infty (-1)^i \frac where ''x'' is any rational number greater or equal to 1. Considering the power series expansion for arctangent, we have : G_x = \arctan\frac. Setting ''x'' = 1 gives the well-known Leibniz formula for pi. Thus, in particular, :\frac=\arctan 1 is a Gregory number. Properties * G_=-(G_x) * \tan(G_x)= \frac See also * Størmer number In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer n for which the greatest prime factor of n^2+1 is greater than or equal to 2n. They are named after Carl Størmer. Sequence The first few Størmer numbers ... References Sets of real numbers {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Set (combinatorics)
In combinatorial mathematics, a large set of positive integers :S = \ is one such that the infinite sum of the reciprocals :\frac+\frac+\frac+\frac+\cdots diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges. Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions. Examples * Every finite subset of the positive integers is small. * The set \ of all positive integers is known to be a large set; this statement is equivalent to the divergence of the harmonic series. More generally, any arithmetic progression (i.e., a set of all integers of the form ''an'' + ''b'' with ''a'' ≥ 1, ''b'' ≥ 1 and ''n'' = 0, 1, 2, 3, ...) is a large set. * The set of square numbers is small (see Basel problem). So is the set of cube numbers, the set of 4th powers, and so on. More generally, the set of positive in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm Of 2
The decimal value of the natural logarithm of 2 is approximately :\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458. The logarithm of 2 in other bases is obtained with the formula :\log_b 2 = \frac. The common logarithm in particular is () :\log_ 2 \approx 0.301\,029\,995\,663\,981\,195. The inverse of this number is the binary logarithm of 10: : \log_2 10 =\frac \approx 3.321\,928\,095 (). By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number. Series representations Rising alternate factorial :\ln 2 = \sum_^\infty \frac=1-\frac12+\frac13-\frac14+\frac15-\frac16+\cdots. This is the well-known "alternating harmonic series". :\ln 2 = \frac +\frac\sum_^\infty \frac. :\ln 2 = \frac +\frac\sum_^\infty \frac. :\ln 2 = \frac +\frac\sum_^\infty \frac. :\ln 2 = \frac +\frac\sum_^\infty \frac. :\ln 2 = \frac +\frac\sum_^\infty \frac. : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Density
In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval as ''n '' grows large. Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this intuition precise for many, but not all, subsets of the naturals (see Schnirelmann density, which is similar to natural density but defined for all subsets of \mathbb). If ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofinite
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum. This use of the prefix "" to describe a property possessed by a set's mplement is consistent with its use in other terms such as " meagre set". Boolean algebras The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, which means that it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the on X. A Boolean algebra A has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra) if and only if there exists an infinite set X such that A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the '' cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an ''infinite set''. For example, the set of all positive integers is infinite: :\. Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set is called finite if there exists a bijection :f\colon S\to\ for some natural number . The number is the set's cardinality, denoted as . The empty set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
