Additive Number Theory

the section of number theory specifically concerned with the expansion of natural numbers 1, 2, 3, . . . into components of definite characteristics. Best known are such problems as expressing a number as a sum of four squares, nine cubes, and so on (known as Waring’s problem) and expressing a number as a sum of a prime number and two squares. A general analytical method worked out by G. Hardy and J. Littlewood in England and I. A. Vinogradov in the USSR is used for the solution of additive types of problems. Many additive number theory problems can be solved by means of elementary methods, particularly by the method of addition of sequences proposed by L. G. Shnirel’man. Iu. V. Linnik has solved a number of problems in this area with the help of the dispersion method developed by him.