in number theory, waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers. for example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. waring's problem was proposed in 1770 by edward waring, after whom it is named. its affirmative answer, known as the hilbertâwaring theorem, was provided by hilbert in 1909. waring's problem has its own mathematics subject classification, 11p05, "waring's problem and variants." ==relationship with lagrange's four-square theorem== long before waring posed his problem, diophantus had asked whether every positive integer could be represented as the sum of four perfect squares greater than or equal to zero. this question later became known as bachet's conjecture, after the 1621 translation of diophantus by claude gaspard bachet de mÃ©ziriac, and it was solved by joseph-louis lagrange in his four-square theorem in 1770, the same year waring made his conjecture. waring sought to generalize this problem by trying to represent all positive integers as the sum of cubes, integers to the fourth power, and so forth, to show that any positive integer may be represented as the sum of other integers raised to a specific exponent, and that there was always a maximum number of integers raised to a certain exponent required to represent all positive integers in this way. ==the number g(k)== for every k, let g(k) denote the minimum number s of kth powers needed to represent all integers. every integer is the sum of one first power, itself, so g(1) = 1. some simple computations show that 7 requires 4 squares, 23 requires 9 cubes, and 79 requires 19 fourth-powers; these examples show that g(2) â¥ 4, g(3) â¥ 9, and g(4) â¥ 19. waring conjectured that these values were in fact the best possible. lagrange's four-square theorem of 1770 states that every natural number is the sum of at most four squares; since three squares are not enough, this theorem establishes g(2) = 4. lagrange's four-square theorem was conjectured in bachet's 1621 edition of diophantus's arithmetica; fermat claimed to have a proof, but did not publish it.{{cite book | last = dickson | first = leonard eugene | authorlink = leonard eugene dickson |
Additive_number_theory Mathematical_problems