Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/9

7. In the first place, $\scriptstyle {n}$ must be odd; otherwise the odd numbers cannot be expressed in this form. Suppose then that $\scriptstyle {n}$ is odd. I shall show that all integers save a finite number can be expressed in the form (7·1); and that the numbers which cannot be so expressed are

   (i) the odd numbers less than $\scriptstyle {n}$,

   (ii) the numbers of the form $\scriptstyle {4^{\lambda }(16\mu +14)}$ less than $\scriptstyle {4n}$,

   (iii) the numbers of the form $\scriptstyle {n+4^{\lambda }(16\mu +14)}$ greater than $\scriptstyle {n}$ and less than $\scriptstyle {9n}$,

   (iv) the numbers of the form

   $\scriptstyle {cn+4^{\kappa }(16\nu +14),\quad (\nu =0,~1,~2,~3,~\ldots )}$,

   greater than $\scriptstyle {9n}$ and less than $\scriptstyle {25n}$, where $\scriptstyle {c=1}$ if $\scriptstyle {n\equiv 1{\pmod {4}}}$, $\scriptstyle {c=9}$ if $\scriptstyle {n\equiv 3{\pmod {4}}}$, $\scriptstyle {\kappa =2}$ if $\scriptstyle {n^{2}\equiv 1{\pmod {16}}}$, and $\scriptstyle {\kappa >2}$ if $\scriptstyle {n^{2}\equiv 9{\pmod {16}}}$.

   First, let us suppose $\scriptstyle {N}$ even. Then, since $\scriptstyle {n}$ is odd and $\scriptstyle {N}$ is even, it is clear that $\scriptstyle {u}$ must be even. Suppose then that

   $\scriptstyle {u=2v,\quad N=2M}$.

   We have to show that $\scriptstyle {M}$ can be expressed in the form

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

   (7·2).

   Since $\scriptstyle {2n\equiv 2{\pmod {4}}}$, it follows from (6·2) that all integers except those which are less than $\scriptstyle {2n}$ and of the form $\scriptstyle {4^{\lambda }(8\mu +7)}$ can be expressed in the form (7·2). Hence the only even integers which cannot be expressed in the form (7·1) are those of the form $\scriptstyle {4^{\lambda }(16\mu +14)}$ less than $\scriptstyle {4n}$.

   This completes the discussion of the case in which $\scriptstyle {N}$ is even. If $\scriptstyle {N}$ is odd the discussion is more difficult. In the first place, all odd numbers less than $\scriptstyle {n}$ are plainly among the exceptions. Secondly, since $\scriptstyle {n}$ and $\scriptstyle {N}$ are both odd, $\scriptstyle {u}$ must also be odd. We can therefore suppose that

   $\scriptstyle {N=n+2M,\quad u^{2}=1+8\Delta }$,

   where $\scriptstyle {\Delta }$ is an integer of the form $\scriptstyle {{\frac {1}{2}}k(k+1)}$, so that $\scriptstyle {\Delta }$ may assume the values $\scriptstyle {0,~1,~3,~6,~\ldots }$. And we have to consider whether $\scriptstyle {n+2M}$ can be expressed in the form

   $\scriptstyle {2(x^{2}+y^{2}+z^{2})+n(1+8\Delta )}$,

   or $\scriptstyle {M}$ in the form

   $\scriptstyle {x^{2}+y^{2}+z^{2}+4n\Delta }$

   (7·3).

   
   

   If $\scriptstyle {M}$ is not of the form $\scriptstyle {4^{\lambda }(8\mu +7)}$, we can take $\scriptstyle {\Delta =0}$. If it is of this form, and less than $\scriptstyle {4n}$, it is plainly an exception. These numbers give rise to the exceptions specified in (iii) of section 7. We may therefore suppose that $\scriptstyle {M}$ is of the form $\scriptstyle {4^{\lambda }(8\mu +7)}$ and greater than $\scriptstyle {4n}$.