Mason–Stothers theorem

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
• 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
• Descartes' theorem for four kissing circles involves sums of squares
• Fermat's theorem on sums of two squares says which integers are sums of two squares.
  • A separate article discusses Proofs of Fermat's theorem on 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.
• 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 contains sums of squares. (similar to Heron's formula)

Algebra and algebraic geometry

• For representing a polynomial as the sum of squares of polynomials, see Polynomial SOS.
  • For computational optimization, see Sum-of-squares optimization.
• For representing a multivariate polynomial that takes only non-negative values over the reals as a sum of squares of rational functions, see Hilbert's seventeenth problem.
• The Brahmagupta–Fibonacci identity says the set of all sums of two squares is closed under multiplication.

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.
• 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

See also

• Sums of powers