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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 ]
Pierre de Fermat gave a criterion for numbers of the form 8a + 1 and 8a + 3 to be sums of a square plus twice another square, but did not provide a proof. [1] N. Beguelin noticed in 1774 [2] that every positive integer which is neither of the form 8n + 7, nor of the form 4n, is the sum of three squares, but did not provide a satisfactory proof. [3]
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:
Goldbach's weak conjecture, every odd number greater than 5 can be expressed as the sum of three primes, is a consequence of Goldbach's conjecture. Ivan Vinogradov proved it for large enough n (Vinogradov's theorem) in 1937, [1] and Harald Helfgott extended this to a full proof of Goldbach's weak conjecture in 2013. [2] [3] [4]
A result of Albrecht Pfister [8] shows that a positive semidefinite form in n variables can be expressed as a sum of 2 n squares. [9] Dubois showed in 1967 that the answer is negative in general for ordered fields. [10] In this case one can say that a positive polynomial is a sum of weighted squares of rational functions with positive ...
Induction step: Given that S(k) holds for some value of k ≥ 12 (induction hypothesis), prove that S(k + 1) holds, too. Assume S(k) is true for some arbitrary k ≥ 12. If there is a solution for k dollars that includes at least one 4-dollar coin, replace it by a 5-dollar coin to make k + 1 dollars.
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.