# Page:On the expression of a number in the form ๐๐ฅยฒ+๐๐ฆยฒ+๐๐งยฒ+๐๐ขยฒ.djvu/8

 where $k=14.4^{\lambda-\kappa-2}-n\nu-\frac{1}{8}(n+7)$
is an integer; and so $N-nu^2$ is not of the form $x^2+y^2+z^2$.

In order to prove (ii) we may suppose, as usual, that

 $N=4^\lambda(8\mu+7)$.
If $\lambda=0$, take $u=1$. Then

 $N-nu^2=8\mu+7-n\equiv 6\pmod{8}$.
If $\lambda\geq 1$, take $u=2^{\lambda-1}$. Then


 $N-nu^2=4^{\lambda-1}(8k+3)$, where $k=4(\mu+1)-\frac{1}{8}(n+7)$.
In either case the proof may be completed as before. Thus the only numbers which cannot be expressed in the form (5ยท2), in this case, are those of the form $8\mu+7$ not exceeding $n$. In other words, there is no exception when $n=1$; $7$ is the only exception when $n=9$; $7$ and $15$ are the only exceptions when $n=17$; $7$, $15$ and $23$ are the only exceptions when $n=25$.

### (6ยท6)โ$n\equiv 4\pmod{32}$.

By arguments similar to those used in (6ยท5), we can show that

(i) if $n\geq 132$, there is an infinity of integers which cannot be expressed in the form (5ยท2);

(ii) if $n$ is equal to $4$, $36$, $68$, or $100$, there is only a finite number of exceptions, namely the numbers of the form $4^\lambda(8\mu+7)$ not exceeding $n$.

### (6ยท7)โ$n\equiv 20\pmod{32}$.

By arguments similar to those used in (6ยท3), we can show that the only numbers which cannot be expressed in the form (5ยท2) are those of the form $4^\lambda(8\mu+7)$ not exceeding $n$, and those of the form $4^2(8\mu+7)$ lying between $n$ and $4n$.

### (6ยท8)โ$n\equiv 12\pmod{16}$.

By arguments similar to those used in (6ยท4), we can show that the only numbers which cannot be expressed in the form (5ยท2) are those of the form $4^\lambda(8\mu+7)$ less than $n$, and those of the form

 $n+4^\kappa(8\mu+7),\quad(\nu=0,1,2,3,\ldots)$,
lying between $n$ and $4n$, where $\kappa=2$ if $n$ is of the form $4(8k+3)$ and $\kappa>2$ if $n$ is of the form $4(8k+7)$.


We have thus completed the discussion of the form (5ยท2), and determined the exceptional values of $N$ precisely whenever they are in finite number.

## 7.

We shall proceed to consider the form

 $2(x^2+y^2+z^2)+nu^2$ (7ยท1).