Pagina:Werkecarlf02gausrich.djvu/50

adeoque ${\textstyle W}$ productum e factoribus

$${\begin{aligned}&w=1+\rho +\rho ^{4}+\rho ^{9}+{\text{etc.}}+\rho ^{(n-1)^{2}}\\&{\mathfrak {A}}={\frac {1+A+A^{4}+A^{9}+{\text{etc.}}+A^{(a-1)^{2}}}{1+\alpha +\alpha ^{4}+\alpha ^{9}+{\text{etc.}}+a^{(a-1)^{2}}}}\\&{\mathfrak {B}}={\frac {1+B+B^{4}+B^{9}+{\text{etc.}}+B^{(b-1)^{2}}}{1+\beta +\beta ^{4}+\beta ^{9}+{\text{etc.}}+\beta ^{(b-1)^{2}}}}\\&{\mathfrak {C}}={\frac {1+C+C^{4}+C^{9}+{\text{etc.}}+C^{(c-1)^{2}}}{1+\gamma +\gamma ^{4}+\gamma ^{9}+{\text{etc.}}+\gamma ^{(c-1)^{2}}}}{\text{ etc.}}\end{aligned}}$$

Iam factor primus ${\textstyle w}$ determinatus est per disquisitiones supra traditas (art. 19); factores reliqui vero ${\textstyle {\mathfrak {A}}}$, ${\textstyle {\mathfrak {B}}}$, ${\textstyle {\mathfrak {C}}}$ etc. prodeunt per formulas artt. 22, 24, quas ut omnia iuncta habeantur, hic denuo colligimus^[1]. Duodecim casus hic sunt distinguendi, scilicet
I. Si ${\textstyle a}$ est numerus primus (impar) ${\textstyle =p}$, vel talis numeri potestas cum exponente impari, atque ${\textstyle k}$ residuum quadraticum ipsius ${\textstyle p}$, erit factor respondens ${\textstyle {\mathfrak {A}}=+1}$.

II. Si manentibus reliquis ${\textstyle k}$ est non-residuum quadraticum ipsius ${\textstyle p}$, erit ${\textstyle {\mathfrak {A}}=-1}$.

III. Si ${\textstyle a}$ est quadratum numeri primi imparis, altiorve eius potestas cum exponente pari, erit ${\textstyle {\mathfrak {A}}=+1}$.

IV. Si ${\textstyle a}$ est ${\textstyle =4}$, aut altior binarii potestas cum exponente pari, simulque ${\textstyle k}$ formae ${\textstyle 4\mu +1}$, erit ${\textstyle {\mathfrak {A}}=+1}$.

V. Si, manentibus reliquis ut in IV, ${\textstyle k}$ est formae ${\textstyle 4\mu +3}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +1}$, erit ${\textstyle {\mathfrak {A}}=-i}$.

VI. Si, manentibus reliquis ut in IV, ${\textstyle k}$ est formae ${\textstyle 4\mu +3}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +3}$, erit ${\textstyle {\mathfrak {A}}=+i}$.

VII. Si ${\textstyle a}$ est ${\textstyle =8}$, aut altior binarii potestas cum exponente impari, atque ${\textstyle k}$ formae ${\textstyle 8\mu +1}$, erit ${\textstyle {\mathfrak {A}}=+1}$.

VIII. Si, manentibus reliquis ut in VII, ${\textstyle k}$ est formae ${\textstyle 8\mu +5}$, erit ${\textstyle {\mathfrak {A}}=-1}$.

IX. Si, manentibus reliquis ut in VII, ${\textstyle k}$ est formae ${\textstyle 8\mu +3}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +1}$, erit ${\textstyle {\mathfrak {A}}=+i}$.

1. Manifesto, quae illic erant ${\textstyle k}$ et ${\textstyle h}$, hic erunt ${\textstyle {\frac {n}{a}}}$ et ${\textstyle k}$ respectu factoris secundi, ${\textstyle {\frac {n}{b}}}$ et ${\textstyle k}$ respectu factoris tertii etc.