Editing Analytic number theory (section)

=== Additive number theory ===
{{main|Additive number theory}}
One of the most important problems in additive number theory is [[Waring's problem]], which asks whether it is possible, for any ''k''&nbsp;≥&nbsp;2, to write any positive integer as the sum of a bounded number of ''k''th powers,

:<math>n=x_1^k+\cdots+x_\ell^k.</math>

The case for squares, ''k''&nbsp;=&nbsp;2, was [[Lagrange's four-square theorem|answered]] by Lagrange in 1770, who proved that every positive integer is the sum of at most four squares. The general case was proved by [[David Hilbert|Hilbert]] in 1909, using algebraic techniques which gave no explicit bounds. An important breakthrough was the application of analytic tools to the problem by [[G. H. Hardy|Hardy]] and [[John Edensor Littlewood|Littlewood]]. These techniques are known as the circle method, and give explicit upper bounds for the function ''G''(''k''), the smallest number of ''k''th powers needed, such as [[Ivan Matveyevich Vinogradov|Vinogradov]]'s bound

:<math>G(k)\leq k(3\log k+11).</math>