Pagina:Werkecarlf02gausrich.djvu/51

XI. Si, manentibus reliquis ut in VII, ${\textstyle k}$ est formae ${\textstyle 8\mu +7}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +1}$ erit ${\textstyle {\mathfrak {A}}=-i}$.

XII. Si, manentibus reliquis ut in VII, ${\textstyle k}$ est formae ${\textstyle 8\mu +7}$, atque ${\textstyle {\frac {n}{a}}}$ formae ${\textstyle 4\mu +3}$, erit ${\textstyle {\mathfrak {A}}=+i}$.

Casum eum, ubi ${\textstyle a=2}$, praeterimus; hic quidem ${\textstyle {\mathfrak {A}}}$ foret ${\textstyle ={\frac {0}{0}}}$ sive indeterminatus, sed tunc semper ${\textstyle W=0}$.

Factores reliqui ${\textstyle {\mathfrak {B}}}$, ${\textstyle {\mathfrak {C}}}$ etc. perinde pendent a ${\textstyle b}$, ${\textstyle c}$ etc., ut ${\textstyle {\mathfrak {A}}}$ ab ${\textstyle a}$, quatenus in illorum determinationem ingrediuntur.

31.

Secundum hanc methodum alteram exemplum primum art. 29 ita se habet:

Factor ${\textstyle w}$ fit ${\textstyle =(1+i)\surd 2520}$
Pro ${\textstyle a=8}$ factor respondens ${\textstyle {\mathfrak {A}}}$ fit, per casum VIII, ${\textstyle =-1}$
Factori ipsius ${\textstyle n}$ secundo ${\textstyle 9}$ respondet factor ${\textstyle +1}$ (per casum III.)
Factori ${\textstyle 5}$ respondet factor ${\textstyle -1}$ (per casum II.)

Factori ${\textstyle 7}$ respondet factor ${\textstyle -1}$ (per casum II.)

Hinc conflatur productum ${\textstyle W=(-1-i)\surd 2520}$, ut in art. 29.

32.

Quum valor ipsius ${\textstyle W}$ per methodos duas determinari possit, quarum altera relationibus numerorum ${\textstyle {\frac {nk}{a}}}$, ${\textstyle {\frac {nk}{b}}}$, ${\textstyle {\frac {nk}{c}}}$ etc. ad numeros ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. innititur, altera vero a relationibus ipsius ${\textstyle k}$ ad numeros ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. pendet, inter omnes has relationes nexus quidam conditionalis intercedere debet, ita ut quaevis e reliquis determinabilis esse debeat. Supponamus, omnes numeros ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. esse numeros primos impares, atque ${\textstyle k}$ accipi ${\textstyle =1}$; distribuanturque factores ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. in duas classes, quarum altera contineat eos, qui sunt formae ${\textstyle 4\mu +1}$, et qui denotentur per ${\textstyle p}$, ${\textstyle p^{\prime }}$, ${\textstyle p^{\prime \prime }}$ etc., altera vero constet ex iis, qui sunt formae ${\textstyle 4\mu +3}$, et qui exprimantur per ${\textstyle q}$, ${\textstyle q^{\prime }}$, ${\textstyle q^{\prime \prime }}$ etc.: multitudinem posteriorum designabimus per ${\textstyle m}$. His ita factis, observamus primo, ${\textstyle n}$ fieri formae ${\textstyle 4\mu +1}$, si ${\textstyle m}$ fuerit par (quorsum etiam referri debet casus is, ubi factores classis alterius omnino desunt, sive ubi ${\textstyle m=0}$), contra ${\textstyle n}$ fieri formae ${\textstyle 4\mu +3}$, si ${\textstyle m}$ fuerit impar. Iam determinatio