Pagina:Werkecarlf02gausrich.djvu/55

$$r^{{\frac {1}{4}}nn}(1+r^{n}+r^{2n}+r^{3n}+{\text{etc.}}+r^{nn-n})$$

quae fit ${\textstyle =nr^{{\frac {1}{4}}nn}}$. In casu priori itaque fit ${\textstyle WW^{\prime }=n}$, in posteriori autem ${\textstyle =n+nr^{{\frac {1}{4}}nn}}$; sed ${\textstyle r^{{\frac {1}{4}}nn}}$ fit ${\textstyle =+1}$, si ${\textstyle n}$ est pariter par, tunc itaque prodit ${\textstyle WW^{\prime }=2n}$; contra fit ${\textstyle r^{{\frac {1}{4}}nn}=-1}$, si ${\textstyle n}$ est impariter par, ubi itaque evadit ${\textstyle WW^{\prime }=0}$. Q. E. D.

36.

Iam per art. 22 constat, si ${\textstyle n}$ sit numerus primus impar, ${\textstyle {\frac {W^{\prime }}{W}}}$ fieri ${\textstyle =+1}$ vel ${\textstyle =-1}$, prout ${\textstyle -1}$ fuerit residuum vel non-residuum ipsius ${\textstyle n}$. Hinc in casu priori esse debebit ${\textstyle W^{2}=+n}$, in posteriori ${\textstyle W^{2}=-n}$; quamobrem per art. 13 concludimus, casum priorem tunc tantum locum habere posse, quando ${\textstyle n}$ sit formae ${\textstyle 4\mu +1}$, casumque posteriorem, quando ${\textstyle n}$ sit formae ${\textstyle 4\mu +3}$.

Denique e combinatione conditionum pro residuis ${\textstyle +2}$ et ${\textstyle -1}$ inventarum sponte sequitur, ${\textstyle -2}$ esse residuum cuiusvis numeri primi formae ${\textstyle 8\mu +1}$ vel ${\textstyle 8\mu +3}$, atque non-residuum cuiusvis numeri primi formae ${\textstyle 8\mu +5}$ vel ${\textstyle 8\mu +7}$.