Pagina:Werkecarlf02gausrich.djvu/54

35.

Numeri primi, quorum residuum vel non-residuum est ${\textstyle -1}$, facile dignoscuntur adiumento theorematis sequentis, quod etiam per se ipsum satis memorabile est.

Theorema. Productum e duobus factoribus

$${\begin{aligned}&W^{\prime }=1+r^{-1}+r^{-4}+{\text{etc.}}+r^{-(n-1)^{2}}\\&W=1+r+r^{4}+{\text{etc.}}+r^{(n-1)^{2}}\end{aligned}}$$

est ${\textstyle =n}$, si ${\textstyle n}$ est impar; vel ${\textstyle =0}$, si ${\textstyle n}$ est impariter par; vel ${\textstyle =2n}$, si ${\textstyle n}$ est pariter par.

Demonstr. Quum manifesto fiat

$${\begin{aligned}W&=r+r^{4}+r^{9}+{\text{etc.}}+r^{nn}\\&=r^{4}+r^{9}+{\text{etc.}}+r^{(n+1)^{2}}\\&=r^{9}+{\text{etc.}}+r^{(n+2)^{2}}{\text{ etc.}}\end{aligned}}$$

productum ${\textstyle W{W}^{\prime }}$ ita quoque exhiberi poterit

$${\begin{aligned}&{\phantom {+,}}1+r+r^{4}+r^{9}+{\text{etc.}}+r^{(n-1)^{2}}\\&+r^{-1}(r+r^{4}+r^{9}+r^{16}+{\text{etc.}}+r^{nn})\\&+r^{-4}(r^{4}+r^{9}+r^{16}+r^{25}+{\text{etc.}}+r^{(n+1)^{2}})\\&+r^{-9}(r^{9}+r^{16}+r^{25}+r^{36}+{\text{etc.}}+r^{(n+2)^{2}})\\&{\text{etc.}}\\&+r^{-(n-1)^{2}}(r^{(n-1)^{2}}+r^{nn}+r^{(n+1)^{2}}+r^{(n+2)^{2}}+{\text{etc.}}+r^{(2n-2)^{2}})\end{aligned}}$$

quod aggregatum verticaliter summatum producit

$${\begin{aligned}&{\phantom {+}}n\\&+r(1+rr+r^{4}+r^{6}+{\text{etc.}}+r^{2n-2})\\&+r^{4}(1+r^{4}+r^{8}+r^{12}+{\text{etc.}}+r^{4n-4})\\&+r^{9}(1+r^{6}+r^{12}+r^{18}+{\text{etc.}}+r^{6n-6})\\&+{\text{etc.}}\\&+r^{(n-1)^{2}}(1+r^{2n-2}+r^{4n-4}+r^{6n-6}+{\text{etc.}}+r^{2(n-1)^{2}})\end{aligned}}$$

Iam si ${\textstyle n}$ impar est, singulae partes huius aggregati, praeter primam ${\textstyle n}$, erunt ${\textstyle =0}$; secunda enim manifesto fit ${\textstyle {\frac {r(1-r^{2n})}{1-rr}}}$, tertia ${\textstyle {\frac {r^{4}(1-r^{4n})}{1-r^{4}}}}$ etc. Quoties vero ${\textstyle n}$ par est, excipere insuper oportebit partem