Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/10

 20 In order to complete the discussion, we must consider the three cases in which $$\scriptstyle{n\equiv 1\pmod{8}}$$, $$\scriptstyle{n\equiv 5\pmod{8}}$$, and $$\scriptstyle{n\equiv 3\pmod{4}}$$ separately. (8·1) $\scriptstyle{n\equiv 1\pmod{8}}$.|undefined

If $$\scriptstyle{\lambda}$$ is equal to $$\scriptstyle{0}$$, $$\scriptstyle{1}$$, or $$\scriptstyle{2}$$, take $$\scriptstyle{\Delta=1}$$. Then$\scriptstyle{M-4n\Delta=4^\lambda(8\mu+7)-4n}$is one of the forms$\scriptstyle{8\nu+3,\quad 4(8\nu+3),\quad 4(8\nu+6)}$.

If $$\scriptstyle{\lambda\geq 3}$$ we cannot take $$\scriptstyle{\Delta=1}$$, since $$\scriptstyle{M-4n\Delta}$$ assumes the form $$\scriptstyle{4(8\nu+7)}$$; so we take $$\scriptstyle{\Delta=3}$$. Then$\scriptstyle{M-4n\Delta=4^\lambda(8\mu+7)-12n}$is of the form $$\scriptstyle{4(8\nu+5)}$$. In either of these cases $$\scriptstyle{M-4n\Delta}$$ is of the form $$\scriptstyle{x^2+y^2+z^2}$$. Hence the only values of $$\scriptstyle{M}$$, other than those already specified, which cannot be expressed in the form (7·3), are those of the form$\scriptstyle{4^\kappa(8\nu+7),\quad(\nu=0,~1,~2,~\ldots,~\kappa>2)}$,lying between $$\scriptstyle{4n}$$ and $$\scriptstyle{12n}$$. In other words, the only numbers greater than $$\scriptstyle{9n}$$ which cannot be expressed in the form (7·1), in this case, are the numbers of the form$\scriptstyle{n+4^\kappa(8\nu+7),\quad(\nu=0,~1,~2,~\ldots,~\kappa>2)}$, lying between $$\scriptstyle{9n}$$ and $$\scriptstyle{25n}$$. (8·2) $\scriptstyle{n\equiv 5\pmod{8}}$.|undefined

If $$\scriptstyle{\lambda\neq 2}$$, take $$\scriptstyle{\Delta=1}$$. Then$\scriptstyle{M-4n\Delta=4^\lambda(8\mu+7)-4n}$is one of the forms$\scriptstyle{8\nu+3,\quad 4(8\nu+2),\quad 4(8\nu+3)}$.

If $$\scriptstyle{\lambda=2}$$, we cannot take $$\scriptstyle{\Delta=1}$$, since $$\scriptstyle{M-4n\Delta}$$ assumes the form $$\scriptstyle{4(8\nu+7)}$$; so we take $$\scriptstyle{\Delta=3}$$. Then$\scriptstyle{M-4n\Delta=4^\lambda(8\mu+7)-12n}$is of the form $$\scriptstyle{4(8\nu+5)}$$. In either of these cases $$\scriptstyle{M-4n\Delta}$$ is of the form $$\scriptstyle{x^2+y^2+z^2}$$. Hence the only values of $$\scriptstyle{M}$$, other than those already specified, which cannot be expressed in the form (7·3), are those of the form $$\scriptstyle{16(8\mu+7)}$$ lying between $$\scriptstyle{4n}$$ and $$\scriptstyle{12n}$$. In other words, the only numbers greater than $$\scriptstyle{9n}$$ which cannot be expressed in the form (7·1), in this case, are the numbers of the form $$\scriptstyle{n+4^2(16\mu+14)}$$ lying between $$\scriptstyle{9n}$$ and $$\scriptstyle{25n}$$.