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

21
in the form ${\displaystyle \scriptstyle {ax^{2}+by^{2}+cz^{2}+du^{2}}}$
1. (8·3) ${\displaystyle \scriptstyle {n\equiv 3{\pmod {4}}}}$.

If ${\displaystyle \scriptstyle {\lambda \neq 1}}$, take ${\displaystyle \scriptstyle {\Delta =1}}$. Then

${\displaystyle \scriptstyle {M-4n\Delta =4^{\lambda }(8\mu +7)-4n}}$

is of one of the forms

${\displaystyle \scriptstyle {8\nu +3,\quad 4(4\nu +1)}}$.

If ${\displaystyle \scriptstyle {\lambda =1}}$, take ${\displaystyle \scriptstyle {\Delta =3}}$. Then

${\displaystyle \scriptstyle {M-4n\Delta =4(8\mu +7)-12n}}$

is of the form ${\displaystyle \scriptstyle {4(4\nu +2)}}$. In either of these cases ${\displaystyle \scriptstyle {M-4n\Delta }}$ is of the form ${\displaystyle \scriptstyle {x^{2}+y^{2}+z^{2}}}$.

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 ${\displaystyle \scriptstyle {M}}$, between ${\displaystyle \scriptstyle {4n}}$ and ${\displaystyle \scriptstyle {12n}}$, for which ${\displaystyle \scriptstyle {\Delta =1}}$ and

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

But the numbers which are multiples of ${\displaystyle \scriptstyle {16}}$ and which cannot be expressed in the form ${\displaystyle \scriptstyle {x^{2}+y^{2}+z^{2}}}$ are the numbers

${\displaystyle \scriptstyle {4^{\kappa }(8\nu +7),\quad (\kappa =2,~3,~4,~\ldots ,\,\nu =0,~1,~2,~\ldots )}}$.

The exceptional values of ${\displaystyle \scriptstyle {M}}$ required are therefore those of the numbers
 ${\displaystyle \scriptstyle {4n+4^{\kappa }(8\nu +7)}}$ (8·31)
which lie between ${\displaystyle \scriptstyle {4n}}$ and ${\displaystyle \scriptstyle {12n}}$ and are of the form
 ${\displaystyle \scriptstyle {4(8\mu +7)}}$ (8·32).
But in order that (8·31) may be of the form (8·32), ${\displaystyle \scriptstyle {\kappa }}$ must be ${\displaystyle \scriptstyle {2}}$ if ${\displaystyle \scriptstyle {n}}$ is of the form ${\displaystyle \scriptstyle {8k+3}}$, and ${\displaystyle \scriptstyle {\kappa }}$ may have any of the values ${\displaystyle \scriptstyle {3,~4,~5,~\ldots }}$ if ${\displaystyle \scriptstyle {n}}$ is of the form ${\displaystyle \scriptstyle {8k+7}}$. It follows that the only numbers greater than ${\displaystyle \scriptstyle {9n}}$ which cannot be expressed in the form (7·1), in this case, are the numbers of the form

${\displaystyle \scriptstyle {9n+4^{\kappa }(16\nu +14),\quad (\nu =0,~1,~2,~\ldots )}}$,

lying between ${\displaystyle \scriptstyle {9n}}$ and ${\displaystyle \scriptstyle {25n}}$, where ${\displaystyle \scriptstyle {\kappa =2}}$ if ${\displaystyle \scriptstyle {n}}$ is of the form ${\displaystyle \scriptstyle {8k+3}}$, and ${\displaystyle \scriptstyle {\kappa >2}}$ if ${\displaystyle \scriptstyle {n}}$ is of the form ${\displaystyle \scriptstyle {8k+7}}$.

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