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

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

   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

   $\scriptstyle {M-4n\Delta =4(8\mu +7-n)\equiv 0{\pmod {16}}}$.

   But 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 numbers

   $\scriptstyle {4n+4^{\kappa }(8\nu +7)}$

   (8·31)

   which lie between $\scriptstyle {4n}$ and $\scriptstyle {12n}$ and are of the form

   $\scriptstyle {4(8\mu +7)}$

   (8·32).

   But 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.

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