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Sum Of Two Squares Theorem
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 Square number, 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 number, prime p \equiv 3 \pmod 4 and k is Parity (mathematics), 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 trip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field Norm
In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Formal definition Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K''. The field ''L'' is then a finite dimensional vector space over ''K''. Multiplication by α, an element of ''L'', :m_\alpha\colon L\to L :m_\alpha (x) = \alpha x, is a ''K''-linear transformation of this vector space into itself. The norm, N''L''/''K''(''α''), is defined as the determinant of this linear transformation. If ''L''/''K'' is a Galois extension, one may compute the norm of α ∈ ''L'' as the product of all the Galois conjugates of α: :\operatorname_(\alpha)=\prod_ \sigma(\alpha), where Gal(''L''/''K'') denotes the Galois group of ''L''/''K''. (Note that there may be a repetition in the terms of the product.) For a general field extension ''L''/''K'', and nonzero α in ''L'', let ''σ''(''α''), ..., σ(''α'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Two principal objects of study are the sumset of two subsets ''A'' and ''B'' of elements from an abelian group ''G'', :A + B = \, and the h-fold sumset of ''A'', :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of ''hA'' from the structure of ''A'': for example, determining which elements can be represented as a sum from ''hA'', where ''A'' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that 2 ... [...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] |
Lagrange's Four-square Theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four. p = a_0^2 + a_1^2 + a_2^2 + a_3^2 where the four numbers a_0, a_1, a_2, a_3 are integers. For illustration, 3, 31, and 310 in several ways, can be represented as the sum of four squares as follows: \begin 3 & = 1^2+1^2+1^2+0^2 \\ pt31 & = 5^2+2^2+1^2+1^2 \\ pt310 & = 17^2+4^2+2^2+1^2 \\ pt& = 16^2 + 7^2 + 2^2 +1^2 \\ pt& = 15^2 + 9^2 + 2^2 +0^2 \\ pt& = 12^2 + 11^2 + 6^2 + 3^2. \end This theorem was proven by Joseph Louis Lagrange in 1770. It is a special case of the Fermat polygonal number theorem. Historical development From examples given in the '' Arithmetica,'' it is clear that Diophantus was aware of the theorem. This book was translated in 1621 into Latin by Bachet (Claude Gaspard Bachet de Méziriac), who stated the theorem in the notes of his translation. Bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Three-square Theorem
In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers :n = x^2 + y^2 + z^2 if and only if is not of the form n = 4^a(8b + 7) for nonnegative integers and . The first numbers that cannot be expressed as the sum of three squares (i.e. numbers that can be expressed as n = 4^a(8b + 7)) are :7, 15, 23, 28, 31, 39, 47, 55, 60, 63, 71 ... . History Pierre de Fermat gave a criterion for numbers of the form 8''a'' + 1 and 8''a'' + 3 to be sums of a square plus twice another square, but did not provide a proof. N. Beguelin noticed in 1774 that every positive integer which is neither of the form 8''n'' + 7, nor of the form 4''n'', is the sum of three squares, but did not provide a satisfactory proof. In 1796 Gauss proved his Eureka theorem that every positive integer ''n'' is the sum of 3 triangular numbers; this is equivalent to the fact that 8''n'' + 3 is a sum o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brahmagupta–Fibonacci Identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says :\begin \left(a^2 + b^2\right)\left(c^2 + d^2\right) & = \left(ac-bd\right)^2 + \left(ad+bc\right)^2 & & (1) \\ & = \left(ac+bd\right)^2 + \left(ad-bc\right)^2. & & (2) \end For example, :(1^2 + 4^2)(2^2 + 7^2) = 26^2 + 15^2 = 30^2 + 1^2. The identity is also known as the Diophantus identity,Daniel Shanks, Solved and unsolved problems in number theory, p.209, American Mathematical Society, Fourth edition 1993. as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity. Brahmagupta proved and used a more general identity (the Brahmagupta identity), equivalent to :\begin \left(a^2 + nb^2\right)\left(c^2 + nd^2\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an expository journal intended for a wide audience of mathematicians, from undergraduate students to research professionals. Articles are chosen on the basis of their broad interest and reviewed and edited for quality of exposition as well as content. In this the ''American Mathematical Monthly'' fulfills a different role from that of typical mathematical research journals. The ''American Mathematical Monthly'' is the most widely read mathematics journal in the world according to records on JSTOR. Tables of contents with article abstracts from 1997–2010 are availablonline The MAA gives the Lester R. Ford Awards annually to "authors of articles of expository excellence" published in the ''American Mathematical Monthly''. Editors *2022– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landau–Ramanujan Constant
In mathematics and the field of number theory, the Landau–Ramanujan constant is the positive real number ''b'' that occurs in a theorem proved by Edmund Landau in 1908, stating that for large x, the number of positive integers below x that are the sum of two square numbers behaves asymptotically as :\dfrac. This constant ''b'' was rediscovered in 1913 by Srinivasa Ramanujan, in the first letter he wrote to G.H. Hardy.S. Ramanujan, letter to G.H. Hardy, 16 January, 1913; see: P. Moree and J. Cazaran, ''On a claim of Ramanujan in his first letter to Hardy'', Exposition. Math. 17 (1999), no.4, 289-311. Sums of two squares By the sum of two squares theorem, the numbers that can be expressed as a sum of two squares of integers are the ones for which each prime number congruent to 3 mod 4 appears with an even exponent in their prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Automorphism group The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2''n'' ''n''!. As a matrix group it is given by the set of all ''n''×''n'' signed permutation matrices. This group is isomorphic to the semidirect product :(\mathbb Z_2)^n \rtimes S_n where the symmetric group ''S''''n'' acts on (Z2)''n'' by permutation (this is a classic example of a wreath product). For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cube, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as \overline). Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (geometry), edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. When the end points both lie on a curve (such as a circle), a line segment is called a chord (geometry), chord (of that curve). In real or complex vector spa ... [...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] |
