Pagina:Werkecarlf02gausrich.djvu/153

72.

Quo theorema generale art. praec. ad numerum ${\textstyle 1+i}$ applicari possit, complexum ${\textstyle C}$ denuo in duos complexus minores ${\textstyle G}$ et ${\textstyle G^{\prime }}$ subdividere oportet, et quidem referemus in complexum ${\textstyle G}$ numeros eos ${\textstyle x+yi}$, pro quibus ${\textstyle ax+by=\xi }$ minor est quam ${\textstyle {\frac {1}{2}}p}$, in alterum ${\textstyle G^{\prime }}$ eos, pro quibus ${\textstyle \xi }$ est maior quam ${\textstyle {\frac {1}{2}}p}$; multitudinem numerorum in complexibus ${\textstyle G}$, ${\textstyle G^{\prime }}$ contentorum resp. per ${\textstyle g}$, ${\textstyle g^{\prime }}$ denotabimus, unde erit ${\textstyle g+g^{\prime }={\frac {1}{4}}(p-1)}$.

Criterium completum numerorum ad ${\textstyle G}$ pertinentium itaque erit, ut tres numeri ${\textstyle \eta }$, ${\textstyle \xi -\eta }$, ${\textstyle p-2\xi }$ sint positivi: nam conditio tertia pro complexu ${\textstyle C}$, secundum quam ${\textstyle p-\xi -\eta }$ positivus esse debet, sub illis implicite iam continetur, quum sit ${\textstyle p-\xi -\eta =(\xi -\eta )+(p-2\xi )}$. Perinde criterium completum numerorum ad ${\textstyle G^{\prime }}$ pertinentium consistet in valoribus positivis trium numerorum ${\textstyle \eta }$, ${\textstyle p-\xi -\eta }$, ${\textstyle 2\xi -p}$.

Hinc facile concluditur, productum cuiusvis numeri complexus ${\textstyle G}$ per numerum ${\textstyle 1+i}$ pertinere ad complexum ${\textstyle C^{\prime \prime \prime }}$; si enim statuitur

$${\begin{aligned}(x+yi)(1+i)&=x^{\prime }+y^{\prime }i,{\text{ atque }}ax^{\prime }+by^{\prime }=\xi ^{\prime },ay^{\prime }-bx^{\prime }=\eta ^{\prime },{\text{ invenitur }}\\\xi ^{\prime }&=\xi -\eta ,\eta ^{\prime }-\xi ^{\prime }=2\eta ,p-\xi ^{\prime }-\eta ^{\prime }=p-2\xi \end{aligned}}$$

i.e. criterium pro numero ${\textstyle x+yi}$ complexui ${\textstyle G}$ subdito identicum est cum criterio pro numero ${\textstyle x^{\prime }+y^{\prime }i}$ ad complexum ${\textstyle C^{\prime \prime \prime }}$ pertinente.

Prorsus simili modo ostenditur, productum cuiusvis numeri complexus ${\textstyle G^{\prime }}$ per ${\textstyle 1+i}$ pertinere ad complexum ${\textstyle C^{\prime \prime }}$.

Erit itaque, si in art. praec. ipsi ${\textstyle k}$ valorem ${\textstyle 1+i}$ tribuimus, ${\textstyle c=0}$, ${\textstyle c^{\prime }=0}$, ${\textstyle c^{\prime \prime }=g^{\prime }}$, ${\textstyle c^{\prime \prime \prime }=g}$, et proin character numeri ${\textstyle 1+i}$ fiet ${\textstyle 3g+2g^{\prime }={\frac {1}{2}}(p-1)+g}$. Et quum characteres numerorum ${\textstyle i}$, ${\textstyle -1}$, sint ${\textstyle {\frac {1}{4}}(p-1)}$, ${\textstyle {\frac {1}{2}}(p-1)}$, characteres numerorum ${\textstyle -1+i}$, ${\textstyle -1-i}$, ${\textstyle 1-i}$ resp. erunt ${\textstyle {\frac {3}{4}}(p-1)+g}$, ${\textstyle g,{\frac {1}{4}}(p-1)+g}$. Totus igitur rei cardo iam in investigatione numeri ${\textstyle g}$ vertitur.

73.

Quae in artt. 69-72 exposuimus, proprie independentia sunt a suppositione, ${\textstyle m}$ esse numerum primarium: abhinc vero saltem supponemus, ${\textstyle a}$ imparem, ${\textstyle b}$ parem esse, praetereaque ${\textstyle a}$, ${\textstyle b}$ et ${\textstyle a-b}$ esse numeros positivos. Ante omnia limites valorum ipsius ${\textstyle x}$ in complexu ${\textstyle G}$ stabilire oportet.