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Schinzel's Theorem
In the geometry of numbers, Schinzel's theorem is the following statement: It was originally proved by and named after Andrzej Schinzel Andrzej Bobola Maria Schinzel (5 April 1937 – 21 August 2021) was a Polish mathematician studying mainly number theory. Education Schinzel received an MSc in 1958 at Warsaw University, Ph.D. in 1960 from Institute of Mathematics of the Pol .... Proof Schinzel proved this theorem by the following construction. If n is an even number, with n=2k, then the circle given by the following equation passes through exactly n points: \left(x-\frac\right)^2 + y^2 = \frac 5^. This circle has radius 5^/2, and is centered at the point (\tfrac12,0). For instance, the figure shows a circle with radius \sqrt 5/2 through four integer points. As an equation of integers, \left(2x-1\right)^2 + (2y)^2 = 5^ is writing 5^ as a sum of two squares, where the first is odd and the second is even. There are exactly 4k ways to write 5^ as a sum of two squares, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry Of Numbers
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by . The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity. Minkowski's results Suppose that \Gamma is a lattice in n-dimensional Euclidean space \mathbb^n and K is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if \operatorname (K)>2^n \operatorname(\mathbb^n/\Gamma), then K contains a nonzero vector in \Gamma. The successive minimum \lambda_k is defined to be the inf of the numbers \lambda such that \lambda K contains k linearly independ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a special ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set \mathbb^2 of pairs of real numbers (the real coordinate plane) augmented by appropriate structure often serves as the canonical example. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called '' Cartesian coordinate system'', a coordinate system that specifies each point uniquely in a plane by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrzej Schinzel
Andrzej Bobola Maria Schinzel (5 April 1937 – 21 August 2021) was a Polish mathematician studying mainly number theory. Education Schinzel received an MSc in 1958 at Warsaw University, Ph.D. in 1960 from Institute of Mathematics of the Polish Academy of Sciences where he studied under Wacław Sierpiński, with a habilitation in 1962. He was a member of the Polish Academy of Sciences. Career Schinzel was a professor at the Institute of Mathematics of the Polish Academy of Sciences (IM PAN). His principal interest was the theory of polynomials. His 1958 conjecture on the prime values of polynomials, known as Schinzel's hypothesis H, both extends the Bunyakovsky conjecture and broadly generalizes the twin prime conjecture. He also proved Schinzel's theorem on the existence of circles through any given number of integer points. Schinzel was the author of over 200 research articles in various branches of number theory, including elementary number theory, elementary, analytic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schinzel Circle
In the geometry of numbers, Schinzel's theorem is the following statement: It was originally proved by and named after Andrzej Schinzel Andrzej Bobola Maria Schinzel (5 April 1937 – 21 August 2021) was a Polish mathematician studying mainly number theory. Education Schinzel received an MSc in 1958 at Warsaw University, Ph.D. in 1960 from Institute of Mathematics of the Pol .... Proof Schinzel proved this theorem by the following construction. If n is an even number, with n=2k, then the circle given by the following equation passes through exactly n points: \left(x-\frac\right)^2 + y^2 = \frac 5^. This circle has radius 5^/2, and is centered at the point (\tfrac12,0). For instance, the figure shows a circle with radius \sqrt 5/2 through four integer points. As an equation of integers, \left(2x-1\right)^2 + (2y)^2 = 5^ is writing 5^ as a sum of two squares, where the first is odd and the second is even. There are exactly 4k ways to write 5^ as a sum of two squares, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Two Squares
In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two squares, such that for some integers , . :''An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no factor , where prime p \equiv 3 \pmod 4 and k is odd.'' In writing a number as a sum of two squares, it is allowed for one of the squares to be zero, or for both of them to be equal to each other, so all squares and all doubles of squares are included in the numbers that can be represented in this way. This theorem supplements Fermat's theorem on sums of two squares which says when a prime number can be written as a sum of two squares, in that it also covers the case for composite numbers. A number may have multiple representations as a sum of two squares, counted by the sum of squares function; for instance, every Pythagorean triple a^2+b^2=c^2 gives a second representation for c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Of Squares Function
In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer as the sum of squares, where representations that differ only in the order of the summands or in the signs of the numbers being squared are counted as different, and is denoted by . Definition The function is defined as :r_k(n) = , \, where , \,\ , denotes the cardinality of a set. In other words, is the number of ways can be written as a sum of squares. For example, r_2(1) = 4 since 1 = 0^2 + (\pm 1)^2 = (\pm 1)^2 + 0^2 where each sum has two sign combinations, and also r_2(2) = 4 since 2 = (\pm 1)^2 + (\pm 1)^2 with four sign combinations. On the other hand, r_2(3) = 0 because there is no way to represent 3 as a sum of two squares. Formulae ''k'' = 2 The number of ways to write a natural number as sum of two squares is given by . It is given explicitly by :r_2(n) = 4(d_1(n)-d_3(n)) where is the number of divisors of wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems About Circles
In mathematics, a theorem is a statement (logic), statement that has been Mathematical proof, proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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