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

6. where

   $\scriptstyle {k=14.4^{\lambda -\kappa -2}-n\nu -{\frac {1}{8}}(n+7)}$

   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

   $\scriptstyle {N-nu^{2}=8\mu +7-n\equiv 6{\pmod {8}}}$.

   If $\scriptstyle {\lambda \geq 1}$, take $\scriptstyle {u=2^{\lambda -1}}$. Then

   $\scriptstyle {N-nu^{2}=4^{\lambda -1}(8k+3)}$,

   where

   $\scriptstyle {k=4(\mu +1)-{\frac {1}{8}}(n+7)}$.

   In either case the proof may be completed as before. Thus the only numbers which cannot be expressed in the form (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) $\scriptstyle {n\equiv 4{\pmod {32}}}$.

   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) $\scriptstyle {n\equiv 20{\pmod {32}}}$.

   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) $\scriptstyle {n\equiv 12{\pmod {16}}}$.

   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.

7. We shall proceed to consider the form

   $\scriptstyle {2(x^{2}+y^{2}+z^{2})+nu^{2}}$

   (7·1).