Waring Problem

a problem in the theory of numbers; formulated (without proof) by the English mathematician E. Waring in 1770. Any integer N ≥1 can be represented in the form of a sum

N = aⁿ₁+ ... + aⁿ_k

of a certain number of terms k, each of which is the nth power of a positive integer; the number of terms k depends only on n. A particular case of the Waring problem is the Lagrange theorem, which concerns the fact that each N is the sum of four squares. The first general solution (for any n) of the Waring problem was given by D. Hilbert (1909) with a very rough estimate of the number of terms k in relation to n. More exact estimates of k were obtained in the 1920’s by G. Hardy and J. Littlewood; and in 1934,1. M. Vinogradov, by the method of trigonometric sums that he created, obtained results that were close to definitive. An elementary solution of the Waring problem was given in 1942 by lu. V. Linnik. The special significance of the Waring problem consists in the fact that during its investigation powerful methods of the analytic theory of numbers were created.