Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/6

5. or is of any of the forms

   $\scriptstyle {4k+2,\quad 4k+3,\quad 8k+5,\quad 16k+12,\quad 32k+20}$

   (5·22),

   then all integers save a finite number, and in fact all integers from $\scriptstyle {4n}$ onwards at any rate, can be expressed in the form (5·2); but that for the remaining values of $\scriptstyle {n}$ there is an infinity of integers which cannot be expressed in the form required.

   In proving the first result we need obviously only consider numbers of the form $\scriptstyle {4^{\lambda }(8\mu +7)}$ greater than $\scriptstyle {n}$, since otherwise we may take $\scriptstyle {u=0}$. The numbers of this form less than $\scriptstyle {n}$ are plainly among the exceptions.
6. I shall consider the various cases which may arise in order of simplicity.

   (6·1) $\scriptstyle {n\equiv 0{\pmod {8}}}$.

   There are an infinity of exceptions. For suppose that

   $\scriptstyle {N=8\mu +7}$.

   Then the number

   $\scriptstyle {N-nu^{2}\equiv 7{\pmod {8}}}$

   cannot be expressed in the form $\scriptstyle {x^{2}+y^{2}+z^{2}}$.

   (6·2) $\scriptstyle {n\equiv 2{\pmod {4}}}$.

   There is only a finite number of exceptions. In proving this we may suppose that $\scriptstyle {N=4^{\lambda }(8\mu +7)}$. Take $\scriptstyle {u=1}$. Then the number

   $\scriptstyle {N-nu^{2}=4^{\lambda }(8\mu +7)-n}$

   is congruent to $\scriptstyle {1}$, $\scriptstyle {2}$, $\scriptstyle {5}$, or $\scriptstyle {6}$ to modulus $\scriptstyle {8}$, and so can be expressed in the form $\scriptstyle {x^{2}+y^{2}+z^{2}}$.

   Hence the only numbers which cannot be expressed in the form (5·2) in this case are the numbers of the form $\scriptstyle {4^{\lambda }(8\mu +7)}$ not exceeding $\scriptstyle {n}$.

   (6·3) $\scriptstyle {n\equiv 5{\pmod {8}}}$.

   There is only a finite number of exceptions. We may suppose again that $\scriptstyle {N=4^{\lambda }(8\mu +7)}$. First, let $\scriptstyle {\lambda \neq 1}$. Take $\scriptstyle {u=1}$. Then

   $\scriptstyle {N-nu^{2}=4^{\lambda }(8\mu +7)-n\equiv 2\ {\mathit {or}}\ 3{\pmod {8}}}$.

   If $\scriptstyle {\lambda =1}$ we cannot take $\scriptstyle {u=1}$, since

   $\scriptstyle {N-n\equiv 7{\pmod {8}}}$;

   so we take $\scriptstyle {u=2}$. Then

   $\scriptstyle {N-nu^{2}=4^{\lambda }(8\mu +7)-4n\equiv 8{\pmod {32}}}$.

   In either of these cases $\scriptstyle {N-nu^{2}}$ is of the form $\scriptstyle {x^{2}+y^{2}+z^{2}}$.

   Hence 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 $\scriptstyle {4(8\mu +7)}$ lying between $\scriptstyle {n}$ and $\scriptstyle {4n}$.