Pagina:Werkecarlf02gausrich.djvu/47

26.

Sit porro ${\textstyle n}$ productum e tribus numeris inter se primis ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$, patetque, si statuatur ${\textstyle bc=b^{\prime }}$, etiam ${\textstyle a}$ et ${\textstyle b^{\prime }}$ inter se primos fore; adeoque ${\textstyle W}$ productum e duobus factoribus

$${\begin{aligned}&1+r^{aa}+r^{4aa}+r^{9aa}+{\text{etc.}}+r^{(b^{\prime }-1)^{2}aa}\\&1+r^{b^{\prime }b^{\prime }}+r^{4b^{\prime }b^{\prime }}+r^{9b^{\prime }b^{\prime }}+{\text{etc.}}+r^{(a-1)^{2}b^{\prime }b^{\prime }}\end{aligned}}$$

Sed quum ${\textstyle r^{aa}}$ sit radix propria aequationis ${\textstyle x^{bc}-1=0}$, erit ipse factor prior productum ex

$${\begin{aligned}&1+\rho ^{bb}+\rho ^{4bb}+\rho ^{9bb}+{\text{etc.}}+\rho ^{(c-1)^{2}bb}\\&1+\rho ^{cc}+\rho ^{4cc}+\rho ^{9cc}+{\text{etc.}}+\rho ^{(b-1)^{2}cc}\end{aligned}}$$

si statuitur ${\textstyle r^{aa}=\rho }$. Hinc patet, ${\textstyle W}$ esse productum e factoribus tribus

$${\begin{aligned}&1+r^{bbcc}+r^{4bbcc}+r^{9bbcc}+{\text{etc.}}+r^{(a-1)^{2}bbcc}\\&1+r^{aacc}+r^{4aacc}+r^{9aacc}+{\text{etc.}}+r^{(b-1)^{2}aacc}\\&1+r^{aabb}+r^{4aabb}+r^{9aabb}+{\text{etc.}}+r^{(c-1)^{2}aabb}\end{aligned}}$$

ubi ${\textstyle r^{bbcc}}$, ${\textstyle r^{aacc}}$, ${\textstyle r^{aabb}}$ erunt resp. radices propriae aequationum ${\textstyle x^{a}-1=0}$, ${\textstyle x^{b}-1=0}$, ${\textstyle x^{c}-1=0}$.

27.

Hinc facile concluditur generaliter, si ${\textstyle n}$ sit productum e factoribus quotcunque inter se primis ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc., ${\textstyle W}$ fieri productum e totidem factoribus, qui sint

$${\begin{aligned}&1+r^{\frac {nn}{aa}}+r^{\frac {4nn}{aa}}+r^{\frac {9nn}{aa}}+{\text{etc.}}+r^{\frac {(a-1)^{2}nn}{aa}}\\&1+r^{\frac {nn}{bb}}+r^{\frac {4nn}{bb}}+r^{\frac {9nn}{bb}}+{\text{etc.}}+r^{\frac {(b-1)^{2}nn}{bb}}\\&1+r^{\frac {nn}{cc}}+r^{\frac {4nn}{cc}}+r^{\frac {9nn}{cc}}+{\text{etc.}}+r^{\frac {(c-1)^{2}nn}{cc}}{\text{ etc.}}\end{aligned}}$$

ubi ${\textstyle r^{\frac {nn}{ab}},r^{\frac {nn}{bb}},r^{\frac {nn}{cc}}}$ etc. erunt radices propriae aequationum ${\textstyle x^{a}-1=0,x^{b}-1=0}$, ${\textstyle x^{c}-1=0}$ etc.

28.

Ex his principiis transitus ad determinationem completam ipsius ${\textstyle W}$ pro valore quocunque ipsius ${\textstyle n}$ sponte iam obvius est. Decomponatur scilicet ${\textstyle n}$ in facto-