Waring's problem

Waring's problem asks for the least number of nonnegative integer kth powers needed to represent every positive integer, for each positive exponent k.

Waring's function: The least number of nonnegative kth powers sufficient to represent every positive integer, as a function of k. It is the quantity Waring's problem asks to determine.
Edward Waring: An English mathematician whose 1770 book stated the conjecture that every positive integer is a sum of a bounded number of like powers. His conjecture gave the problem its name.
Integer power: A number obtained by raising an integer to a nonnegative integer exponent. The allowed summands are precisely kth powers.
Goldbach's conjecture: The conjecture that every even integer greater than two is a sum of two primes. It is a parallel additive problem with primes replacing powers.
Lagrange's four-square theorem: The theorem that every nonnegative integer is a sum of four integer squares. It completely solves the problem for square powers.
Waring's 1770 conjecture: Waring's claim that every positive integer is a sum of a bounded number of kth powers for each positive integer k. It was the original form of the problem, though Waring supplied no proof.
Additive representation: An expression of a number as a sum of elements from a specified set. Waring's problem asks which numbers admit representations using powers.
Vinogradov's theorem: The theorem that every sufficiently large odd integer is a sum of three primes. Its methods and questions resemble those used for sums of powers.
Hilbert's proof of Waring's problem: David Hilbert's proof that every positive exponent has a finite Waring number. It established that a finite universal number of summands exists for every exponent.
Joseph-Louis Lagrange: An eighteenth-century mathematician who proved that every positive integer is a sum of four squares. His theorem confirmed Waring's conjecture in the important square case.

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