Waring's problem

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.