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Mathematical problem in number theory
In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants".

Long before Waring posed his problem, Diophantus had asked whether every positive integer could be represented as the sum of four perfect squares greater than or equal to zero. This question later became known as Bachet's conjecture, after the 1621 translation of Diophantus by Claude Gaspard Bachet de Méziriac, and it was solved by Joseph Louis Lagrange in his four square theorem in 1770, the same year Waring made his conjecture. Waring sought to generalize this problem by trying to represent all positive integers as the sum of cubes, integers to the fourth power, and so forth, to show that any positive integer may be represented as the sum of other integers raised to a specific exponent, and that there was always a maximum number of integers raised to a certain exponent required to represent all positive integers in this way.

From the work of Hardy and Littlewood, the related quantity G(k) was studied with g(k). G(k) 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 positive integers to the power of k. Clearly, G(1) = 1. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that G(2) ≥ 4. Since G(k) ≤ g(k) for all k, this shows that 1=G(2) = 4. Davenport showed that 1=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 1986 and 1989 reduced the 14 biquadrates successively to 13 and 12). The exact value of G(k) is unknown for any other k, but there exist bounds.