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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
One is naturally inclined to ask whether these results are optimal. It turns out that Lagrange's four-square theorem cannot be improved, as there are infinitely many positive integers which are not the sum of three squares. This is because no positive integer which is the sum of three squares can leave a remainder of 7 when divided by 8.
Fermat knew that a fourth power cannot be the sum of two other fourth powers (the n = 4 case of Fermat's Last Theorem; see Fermat's right triangle theorem). Euler conjectured that a fourth power cannot be written as the sum of three fourth powers, but 200 years later, in 1986, this was disproven by Elkies with: 20615673 4 = 18796760 4 ...
Lagrange's four-square theorem can be refined in various ways. For example, Zhi-Wei Sun [14] proved that each natural number can be written as a sum of four squares with some requirements on the choice of these four numbers.
The sum of four cubes problem [1] asks whether every integer is the sum of four cubes of integers. It is conjectured the answer is affirmative, but this conjecture has been neither proven nor disproven. [2] Some of the cubes may be negative numbers, in contrast to Waring's problem on sums of cubes, where they are required to be positive.
While some of Bertinelli’s followers need to “chill out,” she praised “99.9%” of her fans for being “really kind, sweet people who don’t give a flying flip if I have roots or [if] I ...
The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function.It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines.
In mathematics, a sum of radicals is defined as a finite linear combination of n th roots: =, where , are natural numbers and , are real numbers.. A particular special case arising in computational complexity theory is the square-root sum problem, asking whether it is possible to determine the sign of a sum of square roots, with integer coefficients, in polynomial time.