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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 ...
The number of ways to write a natural number as sum of two squares is given by r 2 (n). It is given explicitly by = (() ()) where d 1 (n) is the number of divisors of n which are congruent to 1 modulo 4 and d 3 (n) is the number of divisors of n which are congruent to 3 modulo 4. Using sums, the expression can be written as:
Therefore, the theorem states that it is expressible as the sum of two squares. Indeed, 2450 = 7 2 + 49 2. 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.
Albert Girard was the first to make the observation, characterizing the positive integers (not necessarily primes) that are expressible as the sum of two squares of positive integers; this was published in 1625. [2] [3] The statement that every prime p of the form 4n+1 is the sum of two squares is sometimes called Girard's theorem. [4]
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 squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
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, [1] stating that for large , the number of positive integers below that are the sum of two square numbers behaves asymptotically as
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The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, y i = a + b 1 x 1i + b 2 x 2i + ... + ε i, where y i is the i th observation of the response variable, x ji is the i th observation of the j th ...