# Page:On the expression of a number in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑢².djvu/11

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in the form ${ax^{2}+by^{2}+cz^{2}+du^{2}}$ 1. (8·3) ${n\equiv 3{\pmod {4}}}$ .

If ${\lambda \neq 1}$ , take ${\Delta =1}$ . Then

${M-4n\Delta =4^{\lambda }(8\mu +7)-4n}$ is of one of the forms

${8\nu +3,\quad 4(4\nu +1)}$ .

If ${\lambda =1}$ , take ${\Delta =3}$ . Then

${M-4n\Delta =4(8\mu +7)-12n}$ is of the form ${4(4\nu +2)}$ . In either of these cases ${M-4n\Delta }$ is of the form ${x^{2}+y^{2}+z^{2}}$ .

This completes the proof that there is only a finite number of exceptions. In order to determine what they are in this case, we have to consider the values of ${M}$ , between ${4n}$ and ${12n}$ , for which ${\Delta =1}$ and

${M-4n\Delta =4(8\mu +7-n)\equiv 0{\pmod {16}}}$ .

But the numbers which are multiples of ${16}$ and which cannot be expressed in the form ${x^{2}+y^{2}+z^{2}}$ are the numbers

${4^{\kappa }(8\nu +7),\quad (\kappa =2,~3,~4,~\ldots ,\,\nu =0,~1,~2,~\ldots )}$ .

The exceptional values of ${M}$ required are therefore those of the numbers
 ${4n+4^{\kappa }(8\nu +7)}$ (8·31)
which lie between ${4n}$ and ${12n}$ and are of the form
 ${4(8\mu +7)}$ (8·32).
But in order that (8·31) may be of the form (8·32), ${\kappa }$ must be ${2}$ if ${n}$ is of the form ${8k+3}$ , and ${\kappa }$ may have any of the values ${3,~4,~5,~\ldots }$ if ${n}$ is of the form ${8k+7}$ . It follows that the only numbers greater than ${9n}$ which cannot be expressed in the form (7·1), in this case, are the numbers of the form

${9n+4^{\kappa }(16\nu +14),\quad (\nu =0,~1,~2,~\ldots )}$ ,

lying between ${9n}$ and ${25n}$ , where ${\kappa =2}$ if ${n}$ is of the form ${8k+3}$ , and ${\kappa >2}$ if ${n}$ is of the form ${8k+7}$ .

This completes the proof of the results stated in section 7. 