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Waring's Problem was stated in 1770 by Edward Waring, that every natural number s, greater than two, has an associated number k such that every natural number is the the sum of at most s kth powers of natural numbers. The least such s, when it exists, is denoted by g(k). |
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Waring's Problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated number s such that every natural number is the sum of at most s kth powers of natural numbers. The affirmative answer was provided by David Hilbert. |
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[Lagranges Theorem]? gives the values of g(2)=4 , g(3)=9, and g(4)=19. |
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For every k, we denote the least such s by g(k). |
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David Hilbert proved that g(k) exists for every k. Using the [Hardy Littlewood Method]?, a value for g(k) can be found, except for g(4). |
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[Lagranges Theorem]? states that every natural number is the sum of at least four squares; since three squares are not enough, this theorem establishes g(2)=4. g(3)=9 was established around 1912 and g(4) = 19 in 1986. These values had already been conjectured by Waring. Using the [Hardy Littlewood Method]?, g(k) can now readily be computed for all other values of k as well. |
For every k, we denote the least such s by g(k).
[Lagranges Theorem]? states that every natural number is the sum of at least four squares; since three squares are not enough, this theorem establishes g(2)=4. g(3)=9 was established around 1912 and g(4) = 19 in 1986. These values had already been conjectured by Waring.
Using the [Hardy Littlewood Method]?, g(k) can now readily be computed for all other values of k as well.
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