Page:On the expression of a number in the form ๐๐ฅยฒ+๐๐ฆยฒ+๐๐งยฒ+๐๐ขยฒ.djvu/10

8.

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

(8ยท1)โ$n\equiv 1\pmod{8}$.

If $\lambda$ is equal to $0$, $1$, or $2$, take $\Delta=1$. Then

 $M-4n\Delta=4^\lambda(8\mu+7)-4n$
is one of the forms

 $8\nu+3,\quad 4(8\nu+3),\quad 4(8\nu+6)$.

If $\lambda\geq 3$ we cannot take $\Delta=1$, since $M-4n\Delta$ assumes the form $4(8\nu+7)$; so we take $\Delta=3$. Then

 $M-4n\Delta=4^\lambda(8\nu+7)-12n$
is of the form $4(8\nu+5)$. In either of these cases $M-4n\Delta$ is of the form $x^2+y^2+z^2$. Hence the only values of $M$, other than those already specified, which cannot be expressed in the form (7ยท3), are those of the form

 $4^\kappa(8\nu+7),\quad(\nu=0,1,2,\ldots,\kappa>2)$,
lying between $4n$ and $12n$. In other words, the only numbers greater than $9n$ which cannot be expressed in the form (7ยท1), in this case, are the numbers of the form

 $n+4^\kappa(8\nu+7),\quad(\nu=0,1,2,\ldots,\kappa>2)$,
lying between $9n$ and $25n$.


(8ยท2)โ$n\equiv 5\pmod{8}$.

If $\lambda\neq 2$, take $\Delta=1$. Then

 $M-4n\Delta=4^\lambda(8\mu+7)-4n$
is one of the forms

 $8\nu+3,\quad 4(8\nu+2),\quad 4(8\nu+3)$.

If $\lambda=2$, we cannot take $\Delta=1$, since $M-4n\Delta$ assumes the form $4(8\nu+7)$; so we take $\Delta=3$. Then

 $M-4n\Delta=4^\lambda(8\mu+7)-12n$
is of the form $4(8\nu+5)$. In either of these cases $M-4n\Delta$ is of the form $x^2+y^2+z^2$. Hence the only values of $M$, other than those already specified, which cannot be expressed in the form (7ยท3), are those of the form $16(8\mu+7)$ lying between $4n$ and $12n$. In other words, the only numbers greater than $9n$ which cannot be expressed in the form (7ยท1), in this case, are the numbers of the form $n+4^2(16\mu+14)$ lying between $9n$ and $25n$.