Representations of Integers as Sums of Squares

What do the relations have in common? Obviously, their right hand members are all sums of squares. One way to describe those relations is as follows:

In this book, when we speak of a quadratic form,* we mean a rational, integral quadratic form, unless the contrary is stated explicitly. Given a quadratic form Q, let N _Q be the set of values of Q where x ∈ ℤ ^k (i.e., x _i ∈ ℤ for i = 1, 2, ..., k); clearly, N _Q ⊂ ℤ. If \( Q = \sum\nolimits_{i = 1}^k {x_i^2} \), we denote N _Q by N _k . The main problems that we shall study can now be formulated as follows:

After the discussion of representations as sums of two squares, it appears reasonable to consider next the case of representations as sums of three squares. However, as already indicated, the case of k odd is more difficult than that of k even and, what is more important, requires different methods.

As already remarked in Chapter 3, it is easy to verify the “only if” part of Legendre’s Theorem 3.1, which we recall here in its complete form.

In this chapter we consider a diagonal form more general than a simple sum of squares. We shall be concerned with the ternary quadratic forms Q(x,y,z) = ax² + by² + cz². If a, b, c are positive integers, then strictly speaking this form is also a sum of squares, because it can be written as but the number of squares varies with the values of the coefficients a, b, c. In fact, however, Q is an arbitrary diagonal form, with a, b, c integers, but not necessarily positive.

It is clear that, for k ≥ 4, every nonnegative integer is representable as a sum of k squares. Indeed, one can always write \( n = \sum\nolimits_{{i = 1}}^4 {x_i^2} + {0^2} + \cdot \cdot \cdot + {0^2} \), with an arbitrary number of zeros. On the other hand, the number of representations r _k (n) increases very rapidly with n (see Chapter 12), and besides the representations with k — 4 zeros, one usually finds others with fewer zeros or none at all. For example, if k = 5, then 5 = 2² + 1² + 0² + 0² + 0² = 1² + 1² + 1² + 1² + 1².

In 1948 D. H. Lehmer raised an interesting question [156] and answered it almost completely. The remaining gap has only been filled partially during the more than 30 years that have since passed [20].

We recall from Chapter 1 that the determination of r _k (n) can be reduced to that of the coefficient a _n ^(k) in the Taylor series expansion of the function \( {\left( {\sum\nolimits_{{m = - \infty }}^{\infty } {{x^{{{m^2}}}}} } \right)^k} = {\left( {1 + 2\sum\nolimits_{{m = 1}}^{\infty } {{x^{{{m^2}}}}} } \right)^k} \), because this series, denoted traditionally by |θ₃(x)} ^k , equals \( \sum\nolimits_{{ - \infty < {m_i} < \infty }} {{x^{{m_1^2 + m_2^2 + \cdot \cdot \cdot + m_k^2}}}} = 1 + \sum\nolimits_{{n = 1}}^{\infty } {{r_k}} (n){x^n} \). It follows that if \( {\left\{ {{\theta_3}(x)} \right\}^k} = \sum\nolimits_{{n = 0}}^{\infty } {a_n^{{(k)}}{x^n}} \), then a ₀ ^(k) = 1 for all k, and for n ≥ 1, r _k (n) = a _n ^(k) .

In the previous chapter, we have presented those fundamental properties of the theta functions which will be used in the present one to obtain the number r _k (n) of representations of a natural number n as a sum of an even number k of squares.

In [54] Dickson lists about 70 mathematicians who have made contributions to the problem of representations of integers by sums of five or more squares and to that of relations between the numbers of those representations. It is not possible to quote all these results, but in order to convey the flavor of some of them, we shall mention a few without proofs.

We have presented in Chapters 2, 3, and 4 the contributions made by mathematicians, from antiquity until well into the 19th century, to the problem of representations of natural integers by sums of k = 2, 4, and 3 squares. Theorems were obtained that presented the number r _k (n) of these representations as sums of divisor functions (for k = 2 and 4) or sums over Jacobi symbols (for k = 3). Next, in Chapters 8 and 9, we studied a method developed mainly by Jacobi, based on the use of theta functions, by which one obtains similar results, involving divisor functions, for an even number of squares. The method is entirely successful for k ≤ 8, while for k ≥ 10 the formulae contain, besides sums of divisor functions, also more complicated additional terms. Even for k ≤ 8, the cases k = 5 and k = 7 were bypassed. Nevertheless, as mentioned earlier (see Chapter 10), Eisenstein presented formulae depending (as in the case k = 3) on Jacobi symbols for r₅(n) and formulae for r₅(n) of a different type were proposed and proved by Stieltjes (1856–1894) (see [259], [260]), and by Hurwitz [118]. Eisenstein also stated formulae for r₇(n), expressed, like those for r₅(n) with the help of the Legendre-Jacobi symbol. Proofs of Eisenstein’s formulae were given by Smith [253, 254], who deduced them from his own, of a different appearance, and by Minkowski (1864–1909) (see [178]).

We mentioned in Chapter 1 that the number r _s (n) of solutions of the Diophantine equation is the coefficient of x ⁿ in the Taylor expansion of the function \( 1 + \sum\nolimits_{{n = 1}}^{\infty } {{r_s}(n){x^n}} \). Here, as in Chapter 8, we write θ(x) for θ₃(1;x) and we shall suppress the first entry, which will always be z = 1. From (12.1); it follows, by Cauchy’s theorem, that where, we recall, and 𝓒 is a sufficiently small circle around the origin.

Theorem 12.1 is formulated in [72] as follows:

In the preceding chapters we have discussed the representation of natural integers as sums of squares of integers, and only occasionally (e.g., in Chapters 4 and 5) did particular cases of representations by more general quadratic forms occur.