Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/8

 18 is an integer; and so $$\scriptstyle{N-nu^2}$$ is not of the form $$\scriptstyle{x^2+y^2+z^2}$$.

In order to prove (ii) we may suppose, as usual, that$\scriptstyle{N=4^\lambda(8\mu+7)}$.If $$\scriptstyle{\lambda=0}$$, take $$\scriptstyle{u=1}$$. Then$\scriptstyle{N-nu^2=8\mu+7-n\equiv 6\pmod{8}}$.|undefinedIf $$\scriptstyle{\lambda\geq 1}$$, take $$\scriptstyle{u=2^{\lambda-1}}$$. Then $\scriptstyle{N-nu^2=4^{\lambda-1}(8k+3)}$,In either case the proof may be completed as before. Thus the only numbers which cannot be expressed in the form (5·2), in this case, are those of the form $$\scriptstyle{8\mu+7}$$ not exceeding $$\scriptstyle{n}$$. In other words, there is no exception when $$\scriptstyle{n=1}$$; $$\scriptstyle{7}$$ is the only exception when $$\scriptstyle{n=9}$$; $$\scriptstyle{7}$$ and $$\scriptstyle{15}$$ are the only exceptions when $$\scriptstyle{n=17}$$; $$\scriptstyle{7}$$, $$\scriptstyle{15}$$ and $$\scriptstyle{23}$$ are the only exceptions when $$\scriptstyle{n=25}$$. (6·6) $\scriptstyle{n\equiv 4\pmod{32}}$.|undefined

By arguments similar to those used in (6·5), we can show that (i) if $$\scriptstyle{n\geq 132}$$, there is an infinity of integers which cannot be expressed in the form (5·2);

(ii) if $$\scriptstyle{n}$$ is equal to $$\scriptstyle{4}$$, $$\scriptstyle{36}$$, $$\scriptstyle{68}$$, or $$\scriptstyle{100}$$, there is only a finite number of exceptions, namely the numbers of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$ not exceeding $$\scriptstyle{n}$$. (6·7) $\scriptstyle{n\equiv 20\pmod{32}}$.|undefined

By arguments similar to those used in (6·3), we can show that the only numbers which cannot be expressed in the form (5·2) are those of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$ not exceeding $$\scriptstyle{n}$$, and those of the form $$\scriptstyle{4^2(8\mu+7)}$$ lying between $$\scriptstyle{n}$$ and $$\scriptstyle{4n}$$. (6·8) $\scriptstyle{n\equiv 12\pmod{16}}$.|undefined

By arguments similar to those used in (6·4), we can show that the only numbers which cannot be expressed in the form (5·2) are those of the form $$\scriptstyle{4^\lambda(8\mu+7)}$$ less than $\scriptstyle{n}$, and those of the form$\scriptstyle{n+4^\kappa(8\nu+7),\quad(\nu=0,~1,~2,~3,~\ldots)}$, lying between $$\scriptstyle{n}$$ and $$\scriptstyle{4n}$$, where $$\scriptstyle{\kappa=2}$$ if $$\scriptstyle{n}$$ is of the form $$\scriptstyle{4(8k+3)}$$ and $$\scriptstyle{\kappa>2}$$ if $$\scriptstyle{n}$$ is of the form $$\scriptstyle{4(8k+7)}$$.

We have thus completed the discussion of the form (5·2), and determined the exceptional values of $$\scriptstyle{N}$$ precisely whenever they are in finite number.

We shall proceed to consider the form