Waring's problem

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 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.

The number g ( k ) {displaystyle g(k)}

For every k , let g ( k ) denote the minimum number s of k th 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 ) ≥ 4 , g ( 3 ) ≥ 9 , and g ( 4 ) ≥ 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.

Over the years various bounds were established, using increasingly sophisticated and complex proof techniques. For example, Liouville showed that 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 was established from 1909 to 1912 by Wieferich and A. J. Kempner, g ( 4 ) = 19 in 1986 by R. Balasubramanian, F. Dress, and J.-M. Deshouillers, g ( 5 ) = 37 in 1964 by Chen Jingrun, and g ( 6 ) = 73 in 1940 by Pillai.

Let [ x ] and { x } denote the integral and fractional part of x respectively. Since 2 k [ ( 3 / 2 ) k ] − 1 < 3 k , only 2 k and 1 k can be used to represent this number. The most economical representation requires [ ( 3 / 2 ) k ] − 1 terms of 2 k and 2 k − 1 terms of 1 k . It follows that g ( k ) is at least as large as 2 k + [ ( 3 / 2 ) k ] − 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 have proved that

No value of k is known for which 2 k { ( 3 / 2 ) k } + [ ( 3 / 2 ) k ] > 2 k . Mahler proved there can only be a finite number of such k , and Kubina and Wunderlich have shown that any such k must satisfy k > 471,600,000. Thus it is conjectured that this never happens, that is, g ( k ) = 2 k + [ ( 3 / 2 ) k ] − 2 for every positive integer k .

The first few values of g ( k ) are:

1, 4, 9, 19, 37, 73, 143, 279, 548, 1079, 2132, 4223, 8384, 16673, 33203, 66190, 132055 ... (sequence A002804 in the OEIS).

The number G(k)

From the work of Hardy and Littlewood, the related quantity G(k) was studied with g(k). G(k) is defined to be the least positive integer s such that every sufficiently large integer (i.e. every integer greater than some constant) can be represented as a sum of at most s positive integers to the power of k. Since squares are congruent to 0, 1, or 4 (mod 8), no integer congruent to 7 (mod 8) can be represented as a sum of three squares, implying that G(2) ≥ 4. Since G(k) ≤ g(k) for all k, this shows that G(2) = 4. Davenport showed that G(4) = 16 in 1939, by demonstrating that any sufficiently large number congruent to 1 through 14 mod 16 could be written as a sum of 14 fourth powers (Vaughan in 1985 and 1989 reduced the 14 successively to 13 and 12). The exact value of G(k) is unknown for any other k, but there exist bounds.

Lower bounds for G(k)

The number G(k) is greater than or equal to

2^r+2             if k = 2^r with r ≥ 2, or k = 3 × 2^r; p^r+1             if p is a prime greater than 2 and k = p^r(p − 1); (p^r+1 − 1)/2   if p is a prime greater than 2 and k = p^r(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 k + 1.

Upper bounds for G(k)

G(3) is at least four (since cubes are congruent to 0, 1 or −1 mod 9); for numbers less than 1.3×10⁹, 1290740 is the last to require six cubes, and the number of numbers between N and 2N requiring five cubes drops off with increasing N at sufficient speed to have people believe that G(3) = 4; the largest number now known not to be a sum of four cubes is 7373170279850, and the authors give reasonable arguments there that this may be the largest possible. The upper bound G(3) ≤ 7 is due to Linnik in 1943.

13792 is the largest number to require seventeen fourth powers (Deshouillers, Hennecart and Landreau showed in 2000 that every number between 13793 and 10²⁴⁵ required at most sixteen, and Kawada, Wooley and Deshouillers extended Davenport's 1939 result to show that every number above 10²²⁰ required no more than sixteen). Sixteen fourth powers are always needed to write a number of the form 31·16ⁿ.

617597724 is the last number less than 1.3×10⁹ which requires ten fifth powers, and 51033617 the last number less than 1.3×10⁹ which requires eleven.

The upper bounds on the right with k = 5, ..., 20 are due to Vaughan and Wooley.

Using his improved Hardy-Littlewood method, I. M. Vinogradov published numerous refinements leading to

G ( k ) ≤ k ( 3 log ⁡ k + 11 )

in 1947 and, ultimately,

G ( k ) ≤ k ( 2 log ⁡ k + 2 log ⁡ log ⁡ k + C log ⁡ log ⁡ log ⁡ k )

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 G ( k ) (for k ≥ 400 ):

G ( k ) < 2 k log ⁡ k + 2 k log ⁡ log ⁡ k + 12 k .

Further refinements were obtained by Vaughan [1989].

Wooley then established that for some constant C,

G ( k ) ≤ k log ⁡ k + k log ⁡ log ⁡ k + C k .

Vaughan and Wooley have written a comprehensive survey article.