Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/5

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

   $\scriptstyle {N-du^{2}=4^{\lambda }(8\mu -5)}$.

   In neither of these cases is $\scriptstyle {N-du^{2}}$ of the form $\scriptstyle {4^{\lambda }(8\mu +7)}$, and therefore in either case it can be expressed in the form $\scriptstyle {x^{2}+y^{2}+z^{2}}$.

   Finally, let $\scriptstyle {d=7}$. If $\scriptstyle {\mu }$ is equal to $\scriptstyle {0}$, $\scriptstyle {1}$, or $\scriptstyle {2}$, take $\scriptstyle {u=2^{\lambda }}$. Then $\scriptstyle {N-du^{2}}$ is equal to $\scriptstyle {0}$, $\scriptstyle {2.4^{\lambda +1}}$, or $\scriptstyle {4^{\lambda +2}}$. If $\scriptstyle {\mu \geq 3}$, take $\scriptstyle {u=2^{\lambda +1}}$. Then

   $\scriptstyle {N-du^{2}=4^{\lambda }(8\mu -21)}$.

   Therefore in either case $\scriptstyle {N-du^{2}}$ can be expressed in the form $\scriptstyle {x^{2}+y^{2}+z^{2}}$.

   Thus in all cases $\scriptstyle {N}$ is expressible in the form (4·1). Similarly we can dispose of the remaining cases, with the help of the results stated in §3. Thus in discussing (2·42) we use the theorem that every number not of the form (3·21) can be expressed in the form (3·2). The proofs differ only in detail, and it is not worth while to state them at length.

5. We have seen that all integers without any exception can be expressed in the form

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

   (5·1),

   
   

   when

   $\scriptstyle {m=1,\quad 1\leq n\leq 7}$,

   
   

   and

   $\scriptstyle {m=2,\quad n=1}$.

   
   

   We shall now consider the values of $\scriptstyle {m}$ and $\scriptstyle {n}$ for which all integers with a finite number of exceptions can be expressed in the form (5·1).

   In the first place $\scriptstyle {m}$ must be $\scriptstyle {1}$ or $\scriptstyle {2}$. For, if $\scriptstyle {m>2}$, we can choose an integer $\scriptstyle {\nu }$ so that

   $\scriptstyle {nu^{2}\not \equiv \nu {\pmod {m}}}$

   for all values of $\scriptstyle {u}$. Then

   $\scriptstyle {\frac {(m\mu +\nu )-nu^{2}}{m}}$,

   where $\scriptstyle {\mu }$ is any positive integer, is not an integer; and so $\scriptstyle {m\mu +\nu }$ can certainly not be expressed in the form (5·1).

   We have therefore only to consider the two cases in which $\scriptstyle {m}$ is $\scriptstyle {1}$ or $\scriptstyle {2}$. First let us consider the form

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

   (5·2).

   
   

   I shall show that, when $\scriptstyle {n}$ has any of the values

   $\scriptstyle {1,~4,~9,~17,~25,~36,~68,~100}$

   (5·21),