Sum of squares

Last updated November 19, 2023

In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts:

Statistics

For partitioning of variance, see Partition of sums of squares
For the "sum of squared deviations", see Least squares
For the "sum of squared differences", see Mean squared error
For the "sum of squared error", see Residual sum of squares
For the "sum of squares due to lack of fit", see Lack-of-fit sum of squares
For sums of squares relating to model predictions, see Explained sum of squares
For sums of squares relating to observations, see Total sum of squares
For sums of squared deviations, see Squared deviations from the mean
For modelling involving sums of squares, see Analysis of variance
For modelling involving the multivariate generalisation of sums of squares, see Multivariate analysis of variance

Number theory

For the sum of squares of consecutive integers, see Square pyramidal number
For representing an integer as a sum of squares of 4 integers, see Lagrange's four-square theorem
Legendre's three-square theorem states which numbers can be expressed as the sum of three squares
Jacobi's four-square theorem gives the number of ways that a number can be represented as the sum of four squares.
For the number of representations of a positive integer as a sum of squares of k integers, see Sum of squares function.
Fermat's theorem on sums of two squares says which primes are sums of two squares.
- The sum of two squares theorem generalizes Fermat's theorem to specify which composite numbers are the sums of two squares.
Pythagorean triples are sets of three integers such that the sum of the squares of the first two equals the square of the third.
A Pythagorean prime is a prime that is the sum of two squares; Fermat's theorem on sums of two squares states which primes are Pythagorean primes.
Pythagorean triangles with integer altitude from the hypotenuse have the sum of squares of inverses of the integer legs equal to the square of the inverse of the integer altitude from the hypotenuse.
Pythagorean quadruples are sets of four integers such that the sum of the squares of the first three equals the square of the fourth.
The Basel problem, solved by Euler in terms of $\pi$ , asked for an exact expression for the sum of the squares of the reciprocals of all positive integers.
Rational trigonometry's triple-quad rule and triple-spread rule contain sums of squares, similar to Heron's formula.
Squaring the square is a combinatorial problem of dividing a two-dimensional square with integer side length into smaller such squares.

Algebra, algebraic geometry, and optimization

Polynomial SOS, polynomials that are sums of squares of other polynomials
The Brahmagupta–Fibonacci identity, representing the product of sums of two squares of polynomials as another sum of squares
Hilbert's seventeenth problem on characterizing the polynomials with non-negative values as sums of squares
Sum-of-squares optimization, nonlinear programming with polynomial SOS constraints
The sum of squared dimensions of a finite group's pairwise nonequivalent complex representations is equal to cardinality of that group.

Euclidean geometry and other inner-product spaces

The Pythagorean theorem says that the square on the hypotenuse of a right triangle is equal in area to the sum of the squares on the legs. The sum of squares is not factorable.
The squared Euclidean distance between two points, equal to the sum of squares of the differences between their coordinates
Heron's formula for the area of a triangle can be re-written as using the sums of squares of a triangle's sides (and the sums of the squares of squares)
The British flag theorem for rectangles equates two sums of two squares
The parallelogram law equates the sum of the squares of the four sides to the sum of the squares of the diagonals
Descartes' theorem for four kissing circles involves sums of squares
The sum of the squares of the edges of a rectangular cuboid equals the square of any space diagonal

Related Research Articles

A Pythagorean triple consists of three positive integers $a$ , $b$ , and $c$ , such that $a 2 + b 2 = c 2$ . Such a triple is commonly written $(a, b, c)$ , and a well-known example is $(3, 4, 5)$ . If $(a, b, c)$ is a Pythagorean triple, then so is $(ka, kb, kc)$ for any positive integer $k$ . A primitive Pythagorean triple is one in which $a$ , $b$ and $c$ are coprime (that is, they have no common divisor larger than 1). For example, $(3, 4, 5)$ is a primitive Pythagorean triple whereas $(6, 8, 10)$ is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.

A Heronian tetrahedron is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles . Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are also integers.

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 superscript 2; for instance, the square of 3 may be written as 3², which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x^2 (caret) or x**2 may be used in place of x². The adjective which corresponds to squaring is quadratic.

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.

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 squares.

<span class="mw-page-title-main">Congruum</span> Spacing between equally-spaced square numbers

In number theory, a congruum is the difference between successive square numbers in an arithmetic progression of three squares. That is, if $,, and are three square numbers that are equally spaced apart from each other, then the spacing between them,, is called a congruum.$

In mathematics and statistics, sums of powers occur in a number of contexts:

<span class="mw-page-title-main">Nonhypotenuse number</span>

In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number cannot form the hypotenuse of a right angle triangle with integer sides.

<span class="mw-page-title-main">Pythagorean theorem</span> Relation between sides of a right triangle

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

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. It has several equivalent formulations, one of which was stated in 1225 by Fibonacci. In its geometric forms, it states:

In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers $a$ and $b$ to equal the reciprocal of a third positive integer $c$ :

In geometry, the inverse Pythagorean theorem is as follows:

Pythagorean Triangles is a book on right triangles, the Pythagorean theorem, and Pythagorean triples. It was originally written in the Polish language by Wacław Sierpiński, and published in Warsaw in 1954. Indian mathematician Ambikeshwar Sharma translated it into English, with some added material from Sierpiński, and published it in the Scripta Mathematica Studies series of Yeshiva University in 1962. Dover Books republished the translation in a paperback edition in 2003. There is also a Russian translation of the 1954 edition.

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