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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.
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 ...
In arithmetic and algebra, the fifth power or sursolid [1] of a number n is the result of multiplying five instances of n together: n 5 = n × n × n × n × n. Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its cube. The sequence of fifth powers of integers is:
In 1966, a counterexample to Euler's sum of powers conjecture was found by Leon J. Lander and Thomas R. Parkin for k = 5: [1] 27 5 + 84 5 + 110 5 + 133 5 = 144 5. In subsequent years, further counterexamples were found, including for k = 4. The latter disproved the more specific Euler quartic conjecture, namely that a 4 + b 4 + c 4 = d 4 has no ...
Fermat polygonal number theorem, that every positive integer is a sum of at most n of the n-gonal numbers; Waring–Goldbach problem, the problem of representing numbers as sums of powers of primes; 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 ...
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%
For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a 1 {\displaystyle a_{1}} and the common difference of successive members is d {\displaystyle d} , then the n {\displaystyle n} -th term of the sequence ( a n {\displaystyle a_{n ...
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 .
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