Pagina:Werkecarlf02gausrich.djvu/42

$$W=1+2\sum R^{a}=1+R+R^{4}+R^{9}+{\text{etc.}}+R^{(n-1)^{2}}$$

adeoque ${\textstyle W=+\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle W=+i\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$.

Contra in casu altero, ubi ${\textstyle k}$ est non-residuum ipsius ${\textstyle n}$, erit

$$W=1+2\Sigma R^{b}$$

Hinc quum manifesto omnes ${\textstyle a}$, ${\textstyle b}$ complexum integrum numerorum ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots }$ expleant, adeoque sit

$$\Sigma R^{a}+\Sigma R^{b}=R+R^{2}+R^{3}+{\text{etc.}}+R^{n-1}=-1$$

fiet

$$W=-1-2\Sigma R^{a}=-(1+R+R^{4}+R^{9}+{\text{etc.}}+R^{(n-1)^{2}})$$

adeoque ${\textstyle W=-\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle W=-i\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$.

Hinc itaque colligitur
primo, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle k}$ residuum quadraticum ipsius ${\textstyle n}$,

$$T=+\surd n,\quad U=0$$

secundo, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle k}$ non-residuum ipsius ${\textstyle n}$,

$$T=-\surd n,\quad U=0$$

tertio, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$, atque ${\textstyle k}$ residuum ipsius ${\textstyle n}$,

$$T=0,\quad U=+\surd n$$

quarto, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$, atque ${\textstyle k}$ non-residuum ipsius ${\textstyle n}$,

$$T=0,\quad U=-\surd n$$

21.

Sit secundo ${\textstyle n}$ quadratum altiorve potestas numeri primi imparis ${\textstyle p}$, statuaturque ${\textstyle n=p^{2\chi }q}$, ita ut sit ${\textstyle q}$ vel ${\textstyle =1}$ vel ${\textstyle =p}$. Hic ante omnia observare convenit, si ${\textstyle \lambda }$ sit integer quicunque per ${\textstyle p^{\chi }}$ non divisibilis, fieri