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

8. 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) $\scriptstyle {n\equiv 1{\pmod {8}}}$.

   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) $\scriptstyle {n\equiv 5{\pmod {8}}}$.

   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 $\scriptstyle {25n}$.