Pagina:Werkecarlf02gausrich.djvu/91

Quarta cum secunda fit identica. Adiumento harum aequationum tres incognitarum eliminare licet, quo pacto omnes sedecim iam ad duas reductae sunt.

17.

Ut vero determinationem completam nanciscamur, investigare conveniet multitudinem solutionum congruentiae $$1+\alpha +\beta +\gamma \equiv 0{\pmod {p}}$$ designantibus ${\textstyle \alpha }$, ${\textstyle \beta }$, ${\textstyle \gamma }$ indefinite numeros e complexibus ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$. Manifesto valor ${\textstyle \alpha =p-1}$ non est admissibilis, quum fieri nequeat ${\textstyle \beta +\gamma \equiv 0}$: substituendo itaque pro ${\textstyle \alpha }$ deinceps valores reliquos, prodibunt ${\textstyle h}$, ${\textstyle i}$, ${\textstyle k}$, ${\textstyle l}$ valores ipsius ${\textstyle 1+\alpha }$ ad ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$, ${\textstyle D}$ resp. pertinentes. Pro quovis autem valore dato ipsius ${\textstyle 1+\alpha }$ ad ${\textstyle A}$ pertinente, puta pro ${\textstyle 1+\alpha =\alpha ^{0}}$, congruentia ${\textstyle \alpha ^{0}+\beta +\gamma \equiv 0}$ totidem solutiones admittet, quot congruentia ${\textstyle 1+\beta ^{\prime }+\gamma ^{\prime }\equiv 0}$ (statuendo scilicet ${\textstyle \beta \equiv \alpha ^{0}\beta ^{\prime }}$, ${\textstyle \gamma \equiv \alpha ^{0}\gamma ^{\prime })}$, i.e. solutiones ${\textstyle (12)=m}$. Perinde pro quovis valore dato ipsius ${\textstyle 1+\alpha }$ ad ${\textstyle B}$ pertinente, puta pro ${\textstyle 1+\alpha =\beta ^{0}}$, congruentia ${\textstyle \beta ^{0}+\beta +\gamma \equiv 0}$ totidem solutiones habebit, quot haec ${\textstyle 1+\alpha ^{\prime }+\beta ^{\prime }\equiv 0}$ (scilicet statuendo ${\textstyle \beta \equiv \beta ^{0}\alpha ^{\prime }}$, ${\textstyle \gamma \equiv \beta ^{0}\beta ^{\prime })}$, i.e. solutiones ${\textstyle (01)=i}$. Similiter pro quolibet valore dato ipsius ${\textstyle 1+\alpha }$ ad ${\textstyle C}$ pertinente, puta pro ${\textstyle 1+\alpha =\gamma ^{0}}$, congruentia ${\textstyle \gamma ^{0}+\beta +\gamma \equiv 0}$ totidem modis diversis solvi poterit, quot haec ${\textstyle 1+\delta +\alpha ^{\prime }\equiv 0}$ (nempe statuendo ${\textstyle \beta \equiv \gamma ^{0}\delta ,\gamma \equiv \gamma ^{0}\alpha ^{\prime })}$, i.e. solutionum multitudo erit ${\textstyle (03)=l}$. Denique pro quovis valore dato ipsius ${\textstyle 1+\alpha }$ ad ${\textstyle D}$ pertinente, puta pro ${\textstyle 1+\alpha =\delta ^{0}}$, congruentia ${\textstyle \delta ^{0}+\beta +\gamma \equiv 0}$ totidem solutiones habebit, quot haec ${\textstyle 1+\gamma ^{\prime }+\delta ^{\prime }\equiv 0}$ (statuendo ${\textstyle \beta \equiv \delta ^{0}\gamma ^{\prime }}$, ${\textstyle \gamma \equiv \delta ^{0}\delta ^{\prime }}$), i.e. (23) ${\textstyle =m}$ solutiones. Omnibus itaque collectis, patet, congruentiam ${\textstyle 1+\alpha +\beta +\gamma \equiv 0}$ admittere $$hm+ii+kl+lm$$ solutiones diversas.

Prorsus vero simili modo eruimus, si pro ${\textstyle \beta }$ singuli deinceps numeri complexus ${\textstyle B}$ substituantur, summam ${\textstyle 1+\beta }$ obtinere resp. ${\textstyle (10)}$, ${\textstyle (11)}$, ${\textstyle (12)}$, ${\textstyle (13)}$ sive ${\textstyle i}$, ${\textstyle l}$, ${\textstyle m}$, ${\textstyle m}$ valores ad ${\textstyle A}$, ${\textstyle B}$, ${\textstyle C}$, ${\textstyle D}$ pertinentes, et pro quovis valore dato ipsius ${\textstyle 1+\beta }$ ad hos complexus pertinente, congruentiam ${\textstyle 1+\beta +\alpha +\gamma \equiv 0}$ resp. ${\textstyle (02)}$, ${\textstyle (31)}$, ${\textstyle (20)}$, ${\textstyle (13)}$ sive ${\textstyle k}$, ${\textstyle m}$, ${\textstyle k}$, ${\textstyle m}$ solutiones diversas admittere, ita ut multitudo omnium solutionum fiat