Pagina:Werkecarlf02gausrich.djvu/43

$${\begin{aligned}&r^{\lambda \lambda }+r^{(\lambda +p^{\chi }q)^{2}}+r^{(\lambda +2p^{\chi }q)^{2}}+r^{(\lambda +3p^{\chi }q)^{2}}+{\text{etc.}}+r^{(\lambda +n-p^{\chi }q)^{2}}\\&\quad =r^{\lambda \lambda }\left\{1+r^{2\lambda p^{\chi }q}+r^{4\lambda p^{\chi }q}+r^{6\lambda p^{\chi }q}+{\text{etc.}}+r^{2\lambda (n-p^{\chi }q)}\right\}={\frac {r^{\lambda \lambda }(1-r^{2\lambda n})}{1-r^{2\lambda p^{\chi }q}}}=0\end{aligned}}$$

Hinc facile perspicietur, fieri

$$W=1+r^{p^{2\chi }}+r^{4p^{2\chi }}+r^{9p^{2\chi }}+{\text{etc.}}+r^{(n-p^{\chi })^{2}}$$

Termini enim reliqui progressionis

$$1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}$$

distribui poterunt in ${\textstyle (p^{\chi }-1)q}$ progressiones partiales, quae singulae sint ${\textstyle p^{\chi }}$ terminorum, et per transformationem modo traditam summas evanescentes conficiant.

Hinc colligitur, in casu eo, ubi fit ${\textstyle q=1}$, sive ubi ${\textstyle n}$ est potestas numeri primi cum exponente pari, fieri

$$W=p^{\chi }=+\surd n,{\text{ adeoque }}T=+\surd n,U=0$$

Contra in casu eo, ubi ${\textstyle q=p}$, sive ubi ${\textstyle n}$ est potestas numeri primi cum exponente impari, statuemus ${\textstyle r^{p^{2\chi }}=\rho }$, unde ${\textstyle \rho }$ erit radix propria aequationis ${\textstyle x^{p}-1=0}$, et quidem ${\textstyle \rho =\cos {\frac {k}{p}}360^{\circ }+i\sin {\frac {k}{p}}360^{\circ }}$, ac dein

$$W=1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(p^{\chi +1}-1)^{2}}=p^{\chi }(1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(p-1)^{2}})$$

Sed summa șeriei ${\textstyle 1+p+p^{4}+p^{9}+{\text{etc.}}+p^{(p-1)^{2}}}$ per art. praec. determinatur, unde sponte concluditur, fieri

$${\begin{aligned}&W=\pm {\sqrt {n}}=T,{\text{ si fuerit }}p{\text{ formae }}4\mu +1\\&W=\pm i{\sqrt {n}}=iU,{\text{ si fuerit }}p{\text{ formae }}4\mu +3\end{aligned}}$$

signo positivo vel negativo valente, prout ${\textstyle k}$ fuerit residuum vel non-residuum ipsius ${\textstyle p}$.

22.

Facile quoque ex iis, quae in artt. 20. et 21 exposita sunt, derivatur propositio sequens, quae infra usum notabilem nobis praestabit. Statuatur

$$W^{\prime }=1+r^{h}+r^{4h}+r^{9h}+{\text{etc.}}+r^{h(n-1)^{2}}$$