1)/2 || if p is a prime greater than 2 and k = pr(p − 1)/2; |- | k + 1 || for all integers k greater than 1. |} In the absence of congruence restrictions, a density argument suggests that G(k) should equal . Upper bounds for G(k) G(3) is at least 4 (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3, is the last to require 6 cubes, and the number of numbers between N and 2N requiring 5 cubes drops off with increasing N at sufficient speed to have people believe that ; the largest number now known not to be a sum of 4 cubes is , and the authors give reasonable arguments there that this may be the largest possible. The upper bound is due to Linnik in 1943. (All nonnegative integers require at most 9 cubes, and the largest integers requiring 9, 8, 7, 6 and 5 cubes are conjectured to be 239, 454, 8042, and , respectively.) is the largest number to require 17 fourth powers (Deshouillers, Hennecart and Landreau showed in 2000 that every number between and 10245 required at most 16, and Kawada, Wooley and Deshouillers extended Davenport's 1939 result to show that every number above 10220 required no more than 16). Numbers of the form 31·16n always require 16 fourth powers. is the last number less than 1.3 that requires 10 fifth powers, and is the last number less than 1.3 that requires 11. The upper bounds on the right with are due to Vaughan and Wooley. Using his improved Hardy-Littlewood method, I. M. Vinogradov published numerous refinements leading to in 1947 and, ultimately, for an unspecified constant C and sufficiently large k in 1959. Applying his p-adic form of the Hardy–Littlewood–Ramanujan–Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, Anatolii Alexeevitch Karatsuba obtained (1985) a new estimate of the Hardy function (for ): Further refinements were obtained by Vaughan in 1989. Wooley then established that for some constant C, Vaughan and Wooley have written a comprehensive survey article. See also Fermat polygonal number theorem, that every positive integer is a sum of at most n of the n-gonal numbers Waring–Goldbach problem, the problem of representing numbers as sums of powers of primes Subset sum problem, an algorithmic problem that can be used to