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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. The affirmative answer, known as the Hilbert-Waring theorem, was provided by David Hilbert in 1909, though one minor case remained unsolved until 1944•. Yuri Linnik gave an elementary proof to this theorem.
The number g(k) For every k, we denote the least such s by g(k). 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. 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.• 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• and A. J. Kempner•, g(4) = 19 in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers, g(5) = 37 in 1964 by Chen Jingrun, and g(6) = 73 in 1940 by Pillai. Apart from a certain ambiguity (details can be found in ), all the other values of g are now also known, as a result of work by Dickson, Pillai, Rubugunday and Ivan M. Niven. They obtained g(k) when k7. If one get integers q, k that satisfies , Their formula contains three cases, and it is conjectured that the second and third case, which has been shown to occur at most finitely many times by Mahler, never occurs,giving the values 1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190,132055 ... listed in Sloane's A002804. The number G(k) From the work of Hardy and Littlewood, more fundamental than g(k) turned out to be G(k), which is defined to be the least positive integer s such that every sufficiently large integer (i.e. every integer greater than some constant) can be represented as a sum of at most s kth powers of positive integers. It is easy to see that G(2)≥ 4 since every integer congruent to 7 modulo 8 cannot be represented as a sum of three squares. Since G(k) ≤ g(k) for all k, this shows that G(2) = 4. Davenport showed that G(4) = 16 in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers; Vaughan in 1985 reduced this to 13. The exact value of G(k) is unknown for any other k, but there exist bounds. Lower bounds for G(k) The number G(k) is greater than or equal to if with r ≥ 2, or if p is a prime greater than 2 and ight) if p is a prime greater than 2 and k + 1 for all integers k greater than 1. In the absence of congruence restrictions, a density argument suggests that G(k) should equal k+1. Upper bounds for G(k) The following upper bounds for G(k) are known: k 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 G(k) =< 7 17 21 33 42 50 59 67 76 84 92 100 109 117 125 134 142 G(3) is at least four (since cubes are congruent to 0, 1 or -1 mod 9); 1290740 is the last number less than 1.3e9 to require six cubes, and the number of numbers between N and 2N requiring five cubes drops off with increasing N at sufficient speed to have people believe G(3)=4; the largest number now known not to be a sum of four cubes is 7373170279850 for a proof.) Further reading Notes | ||||||||
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