Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/7

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(6·4) $n\equiv 3\pmod{4}$.

There is only a finite number of exceptions. Take

$N=4^\lambda(8\mu+7)$.

If $\lambda\geq 1$, take $u=1$. Then

$N-nu^2\equiv 1\ \mathit{or}\ 5\pmod{8}$.

If $\lambda=0$, take $u=2$. Then

$N-nu^2\equiv 3\pmod{8}$.

In either case the proof is completed as before.

In order to determine precisely which are the exceptional numbers, we must consider more particularly the numbers between $n$ and $4n$ for which $\lambda=0$. For these $u$ must be $1$, and

$N-nu^2\equiv 0\pmod{4}$.

But the numbers which are multiples of $4$ and which cannot be expressed in the form $x^2+y^2+z^2$ are the numbers

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

The exceptions required are therefore those of the numbers

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

(6·41)

which lie between $n$ and $4n$ and are of the form

$8\mu+7$

(6·42).

Now in order that (6·41) may be of the form (6·42), $\kappa$ must be $1$ if $n$ is of the form $8k+3$ and $\kappa$ may have any of the values $2,3,4,\ldots$ if $n$ is of the form $8k+7$. Thus the only numbers which cannot be expressed in the form (5·2), in this case, are those of the form $4^\lambda(8\mu+7)$ less than $n$ and those of the form

$n+4^\kappa(8\nu+7),\quad(\nu=0,1,2,3,\ldots)$,

lying between $n$ and $4n$, where $\kappa=1$ if $n$ is of the form $8k+3$, and $\kappa>1$ if $n$ is of the form $8k+7$.

(6·5) $n\equiv 1\pmod{8}$.

In this case we have to prove that

(i) if $n\geq 33$, there is an infinity of integers which cannot be expressed in the form (5·2);

(ii) if $n$ is $1$, $9$, $17$, or $25$, there is only a finite number of exceptions.

In order to prove (i) suppose that $N=7.4^\lambda$. Then obviously $u$ cannot be zero. But if $u$ is not zero $u^2$ is always of the form $4^\kappa(8\nu+1)$. Hence

$N-nu^2=7.4^\lambda-n.4^\kappa(8\nu+1)$.

Since $n\geq 33$, $\lambda$ must be greater than or equal to $\kappa+2$, to ensure that the right-hand side shall not be negative. Hence

$N-nu^2=4^\kappa(8k+7)$,