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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.
The equal sum of like powers formula is often abbreviated as (k, m, n). Small examples with = = (related to generalized taxicab number) include 59 4 + 158 4 = 133 4 + 134 4 (known to Euler) and 3 6 + 19 6 + 22 6 = 10 6 + 15 6 + 23 6 (found by K. Subba Rao in 1934).
2 Power series. Toggle Power series subsection. 2.1 Low-order polylogarithms. 2.2 Exponential function. ... 7.2 Sum of reciprocal of factorials. 7.3 Trigonometry and ...
James Waldby, A Table of Fifth Powers equal to a Fifth Power (2009) R. Gerbicz, J.-C. Meyrignac, U. Beckert, All solutions of the Diophantine equation a 6 + b 6 = c 6 + d 6 + e 6 + f 6 + g 6 for a,b,c,d,e,f,g < 250000 found with a distributed Boinc project; EulerNet: Computing Minimal Equal Sums Of Like Powers; Weisstein, Eric W. "Euler's Sum ...
This sum conjecture is also known as Sum Theorem, and it may be expressed as follows: the Riemann zeta value of an integer n ≥ 2 is equal to the sum of all the valid (i.e. with s 1 > 1) MZVs of the partitions of length k and weight n, with 1 ≤ k ≤ n − 1. In formula: [3]
The sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is n. That is, for each term in the expansion, the exponents of the x i must add up to n. [1] [a] In the case m = 2, this statement reduces to that of the binomial theorem. [1]
sum of powers of two including fractions: Image title: Demonstration that the sum of powers of two from zero to a given power, inclusive, is 1 less than the next power of two, whereas the sum of powers of two from minus-infinity to a given power, inclusive, equals the next power of two by CMG Lee. Width: 100%: Height: 100%
Date/Time Thumbnail Dimensions User Comment; current: 13:02, 3 May 2021: 512 × 512 (3 KB): Cmglee {{Information |Description=Demonstration that the sum of powers of two up to a given power is 1 less than the next power of two by CMG Lee. |Source={{own}} |Date= |Author= Cmglee |Permission= |other_versions= }} Category:Powers of two
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