Waring's problem

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In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers (for example, every number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc.). The affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909.[1] Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants."

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The number g(k)

For every k, we denote by g(k) the minimum number s of kth powers needed to represent all integers. Note we have g(1) = 1. Some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth-powers; these examples show that g(2) ≥ 4, g(3) ≥ 9, and g(4) ≥ 19. Waring conjectured that these values were in fact the best possible.

Lagrange's four-square theorem of 1770 states that every natural number is the sum of at most four squares; since three squares are not enough, this theorem establishes g(2) = 4. Lagrange's four-square theorem was conjectured in Bachet's 1621 edition of Diophantus; Fermat claimed to have a proof, but did not publish it.[2]

Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, Liouville showed that g(4) is at most 53. Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers.

That g(3) = 9 was established from 1909 to 1912 by Wieferich[3] and A. J. Kempner[4], g(4) = 19 in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers,[5][6] g(5) = 37 in 1964 by Chen Jingrun, and g(6) = 73 in 1940 by Pillai.[7]