7 Further Remarks 123By the results stated above,g(k)=w(k)fork= 2 , 3 ,4 and this has been shown to
hold also fork=5 by Chen (1964) and fork=6 by Pillai (1940).
Hilbert’s method of proof yielded rather large upper bounds forg(k). A completely
new approach was developed in the 1920’s by Hardy and Littlewood, using their ana-
lytic ‘circle’ method. They showed that, for eachk∈N, there existsΓk∈Nsuch that
every sufficiently large positive integer is a sum of at mostΓkk-th powers. The least
possible value ofΓkis traditionally denoted byG(k). For example,G( 2 )=4, since
no positive integern≡7 mod 8 is a sum of less than four squares. Davenport (1939)
showed thatG( 4 )=16, but these are the only two values ofkfor which todayG(k)
is known exactly.
It is obvious thatG(k)≤g(k), and in factG(k)<g(k)for allk>2. In par-
ticular, Dickson (1939) showed that 23 and 239 are the only positive integers which
require the maximum 9 cubes. Hardy and Littlewood obtained the upper boundG(k)≤
(k− 2 ) 2 k−^1 +5, but this has been repeatedly improved by Hardy and Littlewood them-
selves, Vinogradov and others. For example, Wooley (1992) has shown thatG(k)≤
k(logk+log logk+O( 1 )).
By using the upper bound for G(k)of Vinogradov (1935), it was shown by
Dickson, Pillai and Niven (1936–1944) thatg(k) =w(k)for any givenk > 6,
provided that
( 3 / 2 )k− ( 3 / 2 )k ≤ 1 − ( 3 / 2 )k / 2 k.It is possible that this inequality holds for everyk∈N.Foragivenk, it may be checked
by direct calculation, and Kubina and Wunderlich (1990) have verified in this way that
the inequality holds ifk ≤471600000. Furthermore, using a p-adic extension by
Ridout (1957) of the theorem of Roth (1955) on the approximation of algebraic num-
bers by rationals, Mahler (1957) proved that there existsk 0 ∈Nsuch that the inequality
holds for allk≥k 0. However, the proof does not provide a means of estimatingk 0.
Thus we have the bizarre situation thatG(k)is known for only two values ofk,
thatg(k)is known for a vast number of values ofkand is given by a simple formula,
probably for allk, but the information aboutg(k)is at present derived from informa-
tion aboutG(k). Is it too much to hope that an examination of the numerical data will
reveal some pattern in the fractional parts of( 3 / 2 )k?
7 FurtherRemarks
There are many good introductory books on the theory of numbers, e.g. Davenport [4],
LeVeque [28] and Scholz [41]. More extensive accounts are given in Hardy and
Wright [15], Hua [18], Narkiewicz [33] and Nivenet al.[34].
Historical information is provided by Dickson [5], Smith [42] and Weil [46], as
well as the classics Euclid [11], Gauss [13] and Dirichlet [6]. Gauss’s masterpiece is
quoted here and in the text as ‘D.A.’
The reader is warned that, besides its use in§1, the word ‘lattice’ also has quite a
different mathematical meaning, which will be encountered in Chapter VIII.
The basic theory of divisibility is discussed more thoroughly than in the usual texts
by Stieltjes [43]. For Proposition 6, see Pr ̈ufer [35]. In the theory of groups, Schreier’s