Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/6

 16 or is of any of the formsthen all integers save a finite number, and in fact all integers from $$\scriptstyle{4n}$$ onwards at any rate, can be expressed in the form (5·2); but that for the remaining values of $$\scriptstyle{n}$$ there is an infinity of integers which cannot be expressed in the form required.

In proving the first result we need obviously only consider numbers of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$ greater than $$\scriptstyle{n}$$, since otherwise we may take $$\scriptstyle{u=0}$$. The numbers of this form less than $$\scriptstyle{n}$$ are plainly among the exceptions. I shall consider the various cases which may arise in order of simplicity. (6·1) $\scriptstyle{n\equiv 0\pmod{8}}$.|undefined

There are an infinity of exceptions. For suppose that$\scriptstyle{N=8\mu+7}$.Then the number$\scriptstyle{N-nu^2\equiv 7\pmod{8}}$|undefinedcannot be expressed in the form $$\scriptstyle{x^2+y^2+z^2}$$. (6·2) $\scriptstyle{n\equiv 2\pmod{4}}$.|undefined

There is only a finite number of exceptions. In proving this we may suppose that $$\scriptstyle{N=4^\lambda(8\mu+7)}$$. Take $$\scriptstyle{u=1}$$. Then the number$\scriptstyle{N-nu^2=4^\lambda(8\mu+7)-n}$is congruent to $$\scriptstyle{1}$$, $$\scriptstyle{2}$$, $$\scriptstyle{5}$$, or $$\scriptstyle{6}$$ to modulus $$\scriptstyle{8}$$, and so can be expressed in the form $$\scriptstyle{x^2+y^2+z^2}$$.

Hence the only numbers which cannot be expressed in the form (5·2) in this case are the numbers of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$ not exceeding $$\scriptstyle{n}$$. (6·3) $\scriptstyle{n\equiv 5\pmod{8}}$.|undefined

There is only a finite number of exceptions. We may suppose again that $$\scriptstyle{N=4^\lambda(8\mu+7)}$$. First, let $$\scriptstyle{\lambda\neq 1}$$. Take $$\scriptstyle{u=1}$$. Then$\scriptstyle{N-nu^2=4^\lambda(8\mu+7)-n\equiv 2\ \mathit{or}\ 3\pmod{8}}$.|undefinedIf $$\scriptstyle{\lambda=1}$$ we cannot take $$\scriptstyle{u=1}$$, since$\scriptstyle{N-n\equiv 7\pmod{8}}$;|undefinedso we take $$\scriptstyle{u=2}$$. Then$\scriptstyle{N-nu^2=4^\lambda(8\mu+7)-4n\equiv 8\pmod{32}}$.|undefinedIn either of these cases $$\scriptstyle{N-nu^2}$$ is of the form $$\scriptstyle{x^2+y^2+z^2}$$.

Hence the only numbers which cannot be expressed in the form (5·2) are those of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$ not exceeding $$\scriptstyle{n}$$, and those of the form $$\scriptstyle{4(8\mu+7)}$$ lying between $$\scriptstyle{n}$$ and $$\scriptstyle{4n}$$.