Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/9

 Rh In the first place, $$\scriptstyle{n}$$ must be odd; otherwise the odd numbers cannot be expressed in this form. Suppose then that $$\scriptstyle{n}$$ is odd. I shall show that all integers save a finite number can be expressed in the form (7·1); and that the numbers which cannot be so expressed are (i) the odd numbers less than $$\scriptstyle{n}$$,

(ii) the numbers of the form $$\scriptstyle{4^\lambda(16\mu+14)}$$ less than $$\scriptstyle{4n}$$,

(iii) the numbers of the form $$\scriptstyle{n+4^\lambda(16\mu+14)}$$ greater than $$\scriptstyle{n}$$ and less than $$\scriptstyle{9n}$$,

(iv) the numbers of the form $\scriptstyle{cn+4^\kappa(16\nu+14),\quad(\nu=0,~1,~2,~3,~\ldots)}$, greater than $$\scriptstyle{9n}$$ and less than $$\scriptstyle{25n}$$, where $$\scriptstyle{c=1}$$ if $$\scriptstyle{n\equiv 1\pmod{4}}$$, $$\scriptstyle{c=9}$$ if $$\scriptstyle{n\equiv 3\pmod{4}}$$, $$\scriptstyle{\kappa=2}$$ if $$\scriptstyle{n^2\equiv 1\pmod{16}}$$, and $$\scriptstyle{\kappa>2}$$ if $$\scriptstyle{n^2\equiv 9\pmod{16}}$$. First, let us suppose $$\scriptstyle{N}$$ even. Then, since $$\scriptstyle{n}$$ is odd and $$\scriptstyle{N}$$ is even, it is clear that $$\scriptstyle{u}$$ must be even. Suppose then that$\scriptstyle{u=2v,\quad N=2M}$.We have to show that $$\scriptstyle{M}$$ can be expressed in the formSince $$\scriptstyle{2n\equiv 2\pmod{4}}$$, it follows from (6·2) that all integers except those which are less than $$\scriptstyle{2n}$$ and of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$ can be expressed in the form (7·2). Hence the only even integers which cannot be expressed in the form (7·1) are those of the form $$\scriptstyle{4^\lambda(16\mu+14)}$$ less than $$\scriptstyle{4n}$$.

This completes the discussion of the case in which $$\scriptstyle{N}$$ is even. If $$\scriptstyle{N}$$ is odd the discussion is more difficult. In the first place, all odd numbers less than $$\scriptstyle{n}$$ are plainly among the exceptions. Secondly, since $$\scriptstyle{n}$$ and $$\scriptstyle{N}$$ are both odd, $$\scriptstyle{u}$$ must also be odd. We can therefore suppose that$\scriptstyle{N=n+2M,\quad u^2=1+8\Delta}$,where $$\scriptstyle{\Delta}$$ is an integer of the form $$\scriptstyle{\frac{1}{2}k(k+1)}$$, so that $$\scriptstyle{\Delta}$$ may assume the values $$\scriptstyle{0,~1,~3,~6,~\ldots}$$. And we have to consider whether $$\scriptstyle{n+2M}$$ can be expressed in the form$\scriptstyle{2(x^2+y^2+z^2)+n(1+8\Delta)}$,or $$\scriptstyle{M}$$ in the form

If $$\scriptstyle{M}$$ is not of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$, we can take $$\scriptstyle{\Delta=0}$$. If it is of this form, and less than $$\scriptstyle{4n}$$, it is plainly an exception. These numbers give rise to the exceptions specified in (iii) of section 7. We may therefore suppose that $$\scriptstyle{M}$$ is of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$ and greater than $$\scriptstyle{4n}$$.