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- Squares in number theory

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 if and only if n is not of the form for nonnegative integers a a

Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n

Unit square

In mathematics, a unit square is a square whose sides have length 1. Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points (0, 0), (1, 0), (0,

Gauss's lemma (number theory)

Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of

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

Sums of powers

In mathematics and statistics, sums of powers occur in a number of contexts:
* Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two square

Olga Taussky-Todd

Olga Taussky-Todd (August 30, 1906, Olomouc, Austria-Hungary (present-day Olomouc, Czech Republic) – October 7, 1995, Pasadena, California) was an Austrian and later Czech-American mathematician. She

Zolotarev's lemma

In number theory, Zolotarev's lemma states that the Legendre symbol for an integer a modulo an odd prime number p, where p does not divide a, can be computed as the sign of a permutation: where ε deno

History of the Theory of Numbers

History of the Theory of Numbers is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various a

Ramanujan's ternary quadratic form

In number theory, a branch of mathematics, Ramanujan's ternary quadratic form is the algebraic expression x2 + y2 + 10z2 with integral values for x, y and z. Srinivasa Ramanujan considered this expres

Euler's criterion

In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then Euler

Euler's four-square identity

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares.

Square (algebra)

In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a sup

Integer triangle

An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational length; any such rati

Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pyth

Congruence of squares

In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms.

Waring's problem

In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k.

Brocard's conjecture

In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (pn)2 and (pn+1)2, where pn is the nth prime number, for every n ≥ 2. The conjecture is name

Pythagorean prime

A Pythagorean prime is a prime number of the form . Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squar

Büchi's problem

In number theory, Büchi's problem, also known as the n squares' problem, is an open problem named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer M such

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 o

The Book of Squares

The Book of Squares, (Liber Quadratorum in the original Latin) is a book on algebra by Leonardo Fibonacci, published in 1225. It was dedicated to Frederick II, Holy Roman Emperor. After being brought

Basel problem

The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonh

Ramanujan's sum

In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula where (a, q) = 1 means that a only takes on values coprime to q

Jacobi's four-square theorem

Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares.

Pythagorean quadruple

A Pythagorean quadruple is a tuple of integers a, b, c, and d, such that a2 + b2 + c2 = d2. They are solutions of a Diophantine equation and often only positive integer values are considered. However,

Degen's eight-square identity

In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares.Namely: First discovered by Carl

Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coef

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 n as the sum of k squares, where representations that diff

Dixon's factorization method

In number theory, Dixon's factorization method (also Dixon's random squares method or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method

Congruum

In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares.That is, if , , and (for integers , , and ) are three sq

Fermat's right triangle theorem

Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. I

Square number

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, sinc

Sum of two squares theorem

In number theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares, such that n = a2 + b2 for some integers a, b

Legendre's conjecture

Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between and for every positive integer . The conjecture is one of Landau's problems (1912) on prime number

Fermat's theorem on sums of two squares

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: with x and y integers, if and only if The prime numbers for which this is true are ca

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