Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/11

 Rh (8·3) $\scriptstyle{n\equiv 3\pmod{4}}$.|undefined

If $$\scriptstyle{\lambda\neq 1}$$, take $$\scriptstyle{\Delta=1}$$. Then$\scriptstyle{M-4n\Delta=4^\lambda(8\mu+7)-4n}$is of one of the forms$\scriptstyle{8\nu+3,\quad 4(4\nu+1)}$.If $$\scriptstyle{\lambda=1}$$, take $$\scriptstyle{\Delta=3}$$. Then$\scriptstyle{M-4n\Delta=4(8\mu+7)-12n}$is of the form $$\scriptstyle{4(4\nu+2)}$$. In either of these cases $$\scriptstyle{M-4n\Delta}$$ is of the form $$\scriptstyle{x^2+y^2+z^2}$$.

This completes the proof that there is only a finite number of exceptions. In order to determine what they are in this case, we have to consider the values of $$\scriptstyle{M}$$, between $$\scriptstyle{4n}$$ and $$\scriptstyle{12n}$$, for which $$\scriptstyle{\Delta=1}$$ and

$\scriptstyle{M-4n\Delta=4(8\mu+7-n)\equiv 0\pmod{16}}$.|undefinedBut the numbers which are multiples of $$\scriptstyle{16}$$ and which cannot be expressed in the form $$\scriptstyle{x^2+y^2+z^2}$$ are the numbers $\scriptstyle{4^\kappa(8\nu+7),\quad(\kappa=2,~3,~4,~\ldots,\,\nu=0,~1,~2,~\ldots)}$.

The exceptional values of $$\scriptstyle{M}$$ required are therefore those of the numberswhich lie between $$\scriptstyle{4n}$$ and $$\scriptstyle{12n}$$ and are of the formBut in order that (8·31) may be of the form (8·32), $$\scriptstyle{\kappa}$$ must be $$\scriptstyle{2}$$ if $$\scriptstyle{n}$$ is of the form $$\scriptstyle{8k+3}$$, and $$\scriptstyle{\kappa}$$ may have any of the values $$\scriptstyle{3,~4,~5,~\ldots}$$ if $$\scriptstyle{n}$$ is of the form $$\scriptstyle{8k+7}$$. It follows that 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$\scriptstyle{9n+4^\kappa(16\nu+14),\quad(\nu=0,~1,~2,~\ldots)}$,lying between $$\scriptstyle{9n}$$ and $$\scriptstyle{25n}$$, where $$\scriptstyle{\kappa=2}$$ if $$\scriptstyle{n}$$ is of the form $$\scriptstyle{8k+3}$$, and $$\scriptstyle{\kappa>2}$$ if $$\scriptstyle{n}$$ is of the form $$\scriptstyle{8k+7}$$.

This completes the proof of the results stated in section 7.