Waring's problem

Solved, 1909Arithmetic and number theory

Statement

For every integer exponent $k \ge 1$ , there exists a finite number $g(k)$ such that every positive integer $n \ge 1$ can be expressed as the sum of at most $s \le g(k)$ non-negative $k$ -th powers, $n = x_1^k + x_2^k + \dots + x_s^k$ .

Originally posed by Edward Waring in 1770, the qualitative existence of $g(k)$ for every $k$ (the Hilbert–Waring theorem) was proved by David Hilbert in 1909 using polynomial identities derived from $25$ -fold integrals over the unit sphere. In the 1920s, G. H. Hardy and J. E. Littlewood developed the analytic circle method to give quantitative bounds and study $G(k)$ —the number of $k$ -th powers needed for all sufficiently large integers—later refined by I. M. Vinogradov, Hua Luogeng, and Trevor Wooley.

Solved by, 1909David Hilbert

Because small integers like $2^k \lfloor (3/2)^k \rfloor - 1$ can only be summed using $1^k$ and $2^k$ , J. A. Euler conjectured in 1772 that $g(k) = 2^k + \lfloor (3/2)^k \rfloor - 2$ for every $k$ , which is proved for all known values and holds unless $\{(3/2)^k\} > 1 - (3/4)^k$ . The more subtle asymptotic quantity $G(k)$ (for all sufficiently large integers) is known exactly only for $G(2) = 4$ and $G(4) = 16$ (Davenport, 1939), while even $4 \le G(3) \le 7$ remains open.

References

David Hilbert (1909). Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen (Waringsches Problem) · DOI:10.1007/BF01450405
R. C. Vaughan, Trevor D. Wooley (2002). Waring's problem: a survey

History of the problem

1770Joseph-Louis Lagrange proves the four-square theorem ( $g(2) = 4$ ), and Edward Waring conjectures that $g(3) = 9$ , $g(4) = 19$ , and $g(k) < \infty$ for all $k$ .Joseph-Louis Lagrange
1909David Hilbert proves the existence of $g(k)$ for all $k \ge 1$ ; in the same year Arthur Wieferich (completed by Aubrey Kempner in 1912) proves $g(3) = 9$ .David Hilbert
1920G. H. Hardy and J. E. Littlewood introduce the Hardy–Littlewood circle method, obtaining asymptotic formulas for representations by $k$ -th powers and upper bounds on $G(k)$ .
1938Hua Luogeng refines Weyl's inequality and Vinogradov's mean-value method to establish the classical bound $G(k) \le 2^k + 1$ and study Waring–Goldbach problems.Hua Luogeng
1964Chen Jingrun proves that $g(5) = 37$ , settling the exact value for fifth powers.Chen Jingrun
1986Ramachandran Balasubramanian, Jean-Marc Deshouillers, and François Dress prove $g(4) = 19$ , confirming Waring's original 1770 assertion for fourth powers.