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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.
An interpretation of Plato's number is a solution for k = 3. For this special case of m = 1, some of the known solutions satisfying the proposed constraint with n ≤ k, where terms are positive integers, hence giving a partition of a power into like powers, are: [3] k = 3 3 3 + 4 3 + 5 3 = 6 3 k = 4 95800 4 + 217519 4 + 414560 4 = 422481 4 ...
Faulhaber also knew that if a sum for an odd power is given by = + = + + + + then the sum for the even power just below is given by = = + + (+ + + (+)). Note that the polynomial in parentheses is the derivative of the polynomial above with respect to a .
Toggle Power series subsection. 2.1 Low-order polylogarithms. 2.2 Exponential function. 2.3 Trigonometric, ... Sum of reciprocal of factorials
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem.It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n many k th powers of positive integers is itself a k th power, then n is greater than or equal to k:
Subset sum problem, an algorithmic problem that can be used to find the shortest representation of a given number as a sum of powers; Pollock's conjectures; Sums of three cubes, discusses what numbers are the sum of three not necessarily positive cubes; Sums of four cubes problem, discusses whether every integer is the sum of four cubes of integers
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A sum-free sequence of increasing positive integers is one for which no number is the sum of any subset of the ... [3] The sum of the reciprocals of the powers ...