find the shortest representation of a given number as a sum of powers Sums of three cubes, discusses what numbers are the sum of three not necessarily positive cubes Sums of four cubes problem, discusses whether every rational integer is the sum of four cubes of rational integers Notes References G. I. Arkhipov, V. N. Chubarikov, A. A. Karatsuba, "Trigonometric sums in number theory and analysis". Berlin–New-York: Walter de Gruyter, (2004). G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, "Theory of multiple trigonometric sums". Moscow: Nauka, (1987). Yu. V. Linnik, "An elementary solution of the problem of Waring by Schnirelman's method". Mat. Sb., N. Ser. 12 (54), 225–230 (1943). R. C. Vaughan, "A new iterative method in Waring's problem". Acta Mathematica (162), 1–71 (1989). I. M. Vinogradov, "The method of trigonometrical sums in the theory of numbers". Trav. Inst. Math. Stekloff (23), 109 pp. (1947). I. M. Vinogradov, "On an upper bound for G(n)". Izv. Akad. Nauk SSSR Ser. Mat. (23), 637–642 (1959). I. M. Vinogradov, A. A. Karatsuba, "The method of trigonometric sums in number theory", Proc. Steklov Inst. Math., 168, 3–30 (1986); translation from Trudy Mat. Inst. Steklova, 168, 4–30 (1984). Survey, contains the precise formula for G(k), a simplified version of Hilbert's proof and a wealth of references. Has an elementary proof of the existence of G(k) using Schnirelmann density. Has proofs of Lagrange's theorem, the polygonal number theorem, Hilbert's proof of Waring's conjecture and the Hardy–Littlewood proof of the asymptotic formula for the number of ways to represent N as the sum of s kth powers. Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics (1933) (). Has a proof of the Lagrange theorem, accessible to high-school students. External links Additive number theory Mathematical problems Unsolved problems in number theory Squares in number theory