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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 Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2] Since the problem had withstood the attacks of ...
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)
Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, 2450 = 72 + 492 . The prime decomposition of the number 3430 is 2 · 5 · 7 3. This time, the exponent of 7 in the decomposition is 3, an odd number. So 3430 cannot be written as the sum 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 called Pythagorean primes . For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of ...
The result of fitting a set of data points with a quadratic function Conic fitting a set of points using least-squares approximation. The method of least squares is a parameter estimation method in regression analysis based on minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the ...
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 coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K.
v. t. e. Weighted least squares ( WLS ), also known as weighted linear regression, [1] [2] is a generalization of ordinary least squares and linear regression in which knowledge of the unequal variance of observations ( heteroscedasticity) is incorporated into the regression. WLS is also a specialization of generalized least squares, when all ...
The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares, then the sum of grains on all 64 squares is: 1 + 2 + 4 + 8 + ... and so forth for the 64 squares. The total number of grains can be shown to be 2 64 −1 or 18,446,744,073,709,551,615 (eighteen quintillion ...