Waring's problem

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 natural numbers to the power of k. 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 Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, 'Waring's problem and variants.' 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 natural numbers to the power of k. 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 Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, 'Waring's problem and variants.' 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 Méziriac, 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. For every k {displaystyle k} , let g ( k ) {displaystyle g(k)} denote the minimum number s {displaystyle s} of k {displaystyle k} th powers of naturals needed to represent all positive integers. Every positive integer is the sum of one first power, itself, so g ( 1 ) = 1 {displaystyle 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 ) ≥ 4 {displaystyle g(2)geq 4} , g ( 3 ) ≥ 9 {displaystyle g(3)geq 9} , and g ( 4 ) ≥ 19 {displaystyle g(4)geq 19} . Waring conjectured that these lower bounds were in fact exact values. 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 {displaystyle 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. Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, Liouville showed that g ( 4 ) {displaystyle g(4)} is at most 53. Hardy and Littlewood showed that all sufficiently large numbers are the sum of at most 19 fourth powers. That g ( 3 ) = 9 {displaystyle g(3)=9} was established from 1909 to 1912 by Wieferich and A. J. Kempner, g ( 4 ) = 19 {displaystyle g(4)=19} in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers, g ( 5 ) = 37 {displaystyle g(5)=37} in 1964 by Chen Jingrun, and g ( 6 ) = 73 {displaystyle g(6)=73} in 1940 by Pillai. Let ⌊ x ⌋ {displaystyle lfloor x floor } and { x } {displaystyle {x}} respectively denote the integral and fractional part of a positive real number x {displaystyle x} . Since c = 2 k ⌊ ( 3 / 2 ) k ⌋ − 1 < 3 k {displaystyle c=2^{k}lfloor (3/2)^{k} floor -1<3^{k}} , only 2 k {displaystyle 2^{k}} and 1 k {displaystyle 1^{k}} can be used to represent this number c {displaystyle c} ; the most economical representation requires ⌊ ( 3 / 2 ) k ⌋ − 1 {displaystyle lfloor (3/2)^{k} floor -1} terms of 2 k {displaystyle 2^{k}} and 2 k − 1 {displaystyle 2^{k}-1} terms of 1 k {displaystyle 1^{k}} . It follows that g ( k ) {displaystyle g(k)} is at least as large as 2 k + ⌊ ( 3 / 2 ) k ⌋ − 2 {displaystyle 2^{k}+lfloor (3/2)^{k} floor -2} . This was noted by J. A. Euler, the son of Leonhard Euler, in about 1772. Later work by Dickson, Pillai, Rubugunday, Niven and many others has proved that No value of k {displaystyle k} is known for which 2 k { ( 3 / 2 ) k } + ⌊ ( 3 / 2 ) k ⌋ > 2 k {displaystyle 2^{k}{(3/2)^{k}}+lfloor (3/2)^{k} floor >2^{k}} . Mahler proved there can only be a finite number of such k {displaystyle k} , and Kubina and Wunderlich have shown that any such k {displaystyle k} must satisfy k > {displaystyle k>} 471,600,000. Thus it is conjectured that this never happens, that is, g ( k ) = 2 k + ⌊ ( 3 / 2 ) k ⌋ − 2 {displaystyle g(k)=2^{k}+lfloor (3/2)^{k} floor -2} for every positive integer k {displaystyle k} . The first few values of g ( k ) {displaystyle g(k)} are: