Pagina:Werkecarlf02gausrich.djvu/150

69.

Longe vero difficilius absolvuntur moduli ${\textstyle a+bi}$ tales, pro quibus non est ${\textstyle b=0}$ (numeri quartae speciei art. 34 ), pluresque disquisitiones erunt praemittendae. Normam ${\textstyle aa+bb}$, quae erit numerus primus realis formae ${\textstyle 4n+1}$, designabimus per ${\textstyle p}$.

Denotetur per ${\textstyle {S}}$ complexus omnium residuorum simpliciter minimorum pro modulo ${\textstyle a+bi=m}$, exclusa cifra, ita ut multitudo numerorum in ${\textstyle S}$ contentorum sit ${\textstyle =p-1}$. Designet ${\textstyle x+yi}$ indefinite numerum huius systematis, statuaturque ${\textstyle ax+by=\xi }$, ${\textstyle ay-bx=\eta }$. Erunt itaque ${\textstyle \xi }$, ${\textstyle \eta }$ integri inter limites ${\textstyle 0}$ et ${\textstyle p}$ exclusive contenti: in casu praesente enim, ubi ${\textstyle a}$, ${\textstyle b}$ inter se primi sunt, formulae art. 45, puta ${\textstyle \eta \equiv k\xi }$, ${\textstyle \xi \equiv -k\eta {\pmod {p}}}$ docent, neutrum numerorum ${\textstyle \xi }$, ${\textstyle \eta }$ esse posse ${\textstyle =0}$, nisi alter simul evanescat, adeoque fiat ${\textstyle x=0}$, ${\textstyle y=0}$, quam combinationem iam eiecimus. Criterium itaque numeri ${\textstyle x+yi}$ in ${\textstyle S}$ contenti, consistit in eo, ut quatuor numeri ${\textstyle \xi }$, ${\textstyle \eta }$, ${\textstyle p-\xi }$, ${\textstyle p-\eta }$ sint positivi.

Praeterea observamus pro nullo tali numero esse posse ${\textstyle \xi =\eta }$; hinc enim sequeretur ${\textstyle p(x+y)=a(\xi +\eta )+b(\xi -\eta )=2a\xi }$, quod est absurdum, quum nullus factorum ${\textstyle 2}$, ${\textstyle a}$, ${\textstyle \xi }$ per ${\textstyle p}$ divisibilis sit. Simili ratione aequatio ${\textstyle p(x-y+a+b)=2a\xi +(a+b)(p-\xi -\eta )}$ docet, esse non posse ${\textstyle \xi +\eta =p}$. Quapropter quum numeri ${\textstyle \xi -\eta }$, ${\textstyle p-\xi -\eta }$ esse debeant vel positivi vel negativi, hinc petimus subdivisionem systematis ${\textstyle S}$ in quatuor complexus ${\textstyle C}$, ${\textstyle C^{\prime }}$, ${\textstyle C^{\prime \prime }}$, ${\textstyle C^{\prime \prime \prime }}$, puta ut coniiciantur

$${\begin{array}{c|c}{\text{in complexum}}&{\text{numeri pro quibus}}\\\hline C&\xi -\eta {\text{ positivus, }}p-\xi -\eta {\text{ positivus }}\\C^{\prime }&\xi -\eta {\text{ positivus, }}p-\xi -\eta {\text{ negativus }}\\C^{\prime \prime }&\xi -\eta {\text{ negativus, }}p-\xi -\eta {\text{ negativus }}\\C^{\prime \prime \prime }&\xi -\eta {\text{ negativus, }}p-\xi -\eta {\text{ positivus }}\\\end{array}}$$

Criterium itaque numeri complexus ${\textstyle C}$ proprie sextuplex est, puta sex numeri ${\textstyle \xi }$, ${\textstyle \eta }$, ${\textstyle p-\xi }$, ${\textstyle p-\eta }$, ${\textstyle \xi -\eta }$, ${\textstyle p-\xi -\eta }$ positivi esse debent; sed manifesto conditiones 2, 5 et 6 iam sponte implicant reliquas. Similia circa complexus ${\textstyle C^{\prime }}$, ${\textstyle C^{\prime \prime }}$, ${\textstyle C^{\prime \prime \prime }}$ valent, ita ut criteria completa sint triplicia, puta