Pagina:Werkecarlf02gausrich.djvu/44

denotante ${\textstyle h}$ integrum quemcunque per ${\textstyle p}$ non divisibilem, eritque in casu eo, ubi ${\textstyle n=p}$, vel ubi ${\textstyle n}$ est potestas ipsius ${\textstyle p}$ cum exponente impari, $${\begin{aligned}&W^{\prime }=W,{\text{ si fuerit }}h{\text{ residuum quadraticum ipsius }}p\\&W^{\prime }=-W,{\text{ si fuerit }}h{\text{ non-residuum quadraticum ipsius }}p\end{aligned}}$$ Patet enim, ${\textstyle W^{\prime }}$ oriri ex ${\textstyle W}$, si pro ${\textstyle k}$ substituatur ${\textstyle kh}$; in casu priori autem ${\textstyle k}$ et ${\textstyle kh}$ similes erunt, in posteriori dissimiles, quatenus sunt residua vel non-residua ipsius ${\textstyle p}$.

In casu eo autem, ubi ${\textstyle n}$ est potestas ipsius ${\textstyle p}$ cum exponente pari, manifesto fit ${\textstyle W^{\prime }=+\surd n}$, adeoque semper ${\textstyle W^{\prime }=W}$.

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In artt. 20, 21, 22 consideravimus numeros primos impares, taliumque potestates: superest itaque casus, ubi ${\textstyle n}$ est potestas binarii.

Pro ${\textstyle n=2}$ manifesto fit ${\textstyle W=1+r=0}$.

Pro ${\textstyle n=4}$ prodit ${\textstyle W=1+r+r^{4}+r^{9}=2+2r}$: hinc ${\textstyle W=2+2i}$, quoties ${\textstyle k}$ est formae ${\textstyle 4\mu +1}$, atque ${\textstyle W=2-2i}$, quoties ${\textstyle k}$ est formae ${\textstyle 4\mu +3}$.

Pro ${\textstyle n=8}$ habemus ${\textstyle W=1+r+r^{4}+r^{9}+r^{16}+r^{25}+r^{36}+r^{49}=2+4r+2r^{4}}$ ${\textstyle =4r}$. Hinc erit $${\begin{aligned}&W=(1+i)\surd 8,{\text{ quoties }}k{\text{ est formae }}8\mu +1\\&W=(-1+i)\surd 8,{\text{ quoties }}k{\text{ est formae }}8\mu +3\\&W=(-1-i)\surd 8,{\text{ quoties }}k{\text{ est formae }}8\mu +5\\&W=(1-i)\surd 8,{\text{ quoties }}k{\text{ est formae }}8\mu +7\end{aligned}}$$

Si ${\textstyle n}$ est altior potestas binarii, statuamus ${\textstyle n=2^{2\chi }q}$, ita ut ${\textstyle q}$ sit vel ${\textstyle =1}$ vel ${\textstyle =2}$, atque ${\textstyle \chi }$ maior quam ${\textstyle 1}$. Hic ante omnia observari debet, si ${\textstyle \lambda }$ sit integer quicunque per ${\textstyle 2^{\chi -1}}$ non divisibilis, fieri $${\begin{aligned}&r^{\lambda \lambda }+r^{(\lambda +2^{\chi }q)^{2}}+r^{(\lambda +2.2^{\chi }q)^{2}}+r^{(\lambda +3.2^{\chi }q)^{2}}+{\text{etc.}}+r^{(\lambda +n-2^{\chi }q)^{2}}\\&\quad =r^{\lambda \lambda }\left\{1+r^{2^{\chi +1}\lambda q}+r^{2.2^{\chi +1}\lambda q}+r^{3.2^{\chi +1}\lambda q}+{\text{etc.}}+r^{(2n-2^{\chi +1}q)\lambda }\right\}={\frac {r^{\lambda \lambda }(1-r^{2\lambda n})}{1-r^{2^{\chi +1}\lambda q}}}=0\end{aligned}}$$ Hinc facile perspicietur, fieri $$W=1+r^{2^{2\chi -2}}+r^{4.2^{2\chi -2}}+r^{9.2^{2\chi -2}}+{\text{etc.}}+r^{(n-2^{\chi -1})^{2}}$$