Pagina:Werkecarlf02gausrich.djvu/74

$$\varphi (379,206)=19467-8858+1376-64-5356+3741-680+40=9666$$

quapropter 103 est residuum quadraticum numeri 379.

6.

Quum ad decidendam relationem numeri ${\textstyle b}$ ad ${\textstyle a}$ non opus sit, singulas partes aggregati ${\textstyle \varphi (a,b)}$ computare, sed sufficiat novisse, quot inter eas sint impares, regula nostra ita quoque exhiberi potest:

Fiat ut supra ${\textstyle a=\beta b+c}$, ${\textstyle b=\gamma c+d}$, ${\textstyle c=\delta d+e}$ etc., donec in serie numerorum ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$, ${\textstyle d}$, ${\textstyle e}$ etc. ad unitatem perventum sit. Statuatur ${\textstyle \left[{\frac {1}{2}}a\right]=a^{\prime }}$, ${\textstyle \left[{\frac {1}{2}}b\right]=b^{\prime }}$, ${\textstyle \left[{\frac {1}{2}}c\right]=c^{\prime }}$ etc., sitque ${\textstyle \mu }$ multitudo numerorum imparium in serie ${\textstyle a^{\prime }}$, ${\textstyle b^{\prime }}$, ${\textstyle c^{\prime }}$ etc. eorum, quos immediate sequitur impar; sit porro ${\textstyle \nu }$ multitudo numerorum imparium in serie ${\textstyle \beta }$, ${\textstyle \gamma }$, ${\textstyle \delta }$ etc. eorum, quibus in serie ${\textstyle b^{\prime }}$, ${\textstyle c^{\prime }}$, ${\textstyle d^{\prime }}$ etc. resp. respondet numerus formae ${\textstyle 4n+1}$ vel formae ${\textstyle 4n+2}$. His ita factis, erit ${\textstyle b}$ residuum quadraticum vel non-residuum ipsius ${\textstyle a}$, prout ${\textstyle \mu +\nu }$ est par vel impar, unico casu excepto, ubi simul est ${\textstyle b}$ par atque ${\textstyle a}$ vel formae ${\textstyle 8n+3}$ vel ${\textstyle 8n+5}$, pro quo regula opposita valet.

In exemplo nostro series ${\textstyle a^{\prime }}$, ${\textstyle b^{\prime }}$, ${\textstyle c^{\prime }}$, ${\textstyle d^{\prime }}$, ${\textstyle e^{\prime }}$ duas successiones imparium sistit, unde ${\textstyle \mu =2}$; in serie ${\textstyle b^{\prime }}$, ${\textstyle \gamma ^{\prime }}$, ${\textstyle \delta ^{\prime }}$, ${\textstyle \varepsilon ^{\prime }}$, duo quidem impares adsunt, sed quibus in serie ${\textstyle b^{\prime }}$, ${\textstyle c^{\prime }}$, ${\textstyle d^{\prime }}$, ${\textstyle e^{\prime }}$ respondent numeri formae ${\textstyle 4n+3}$, unde ${\textstyle \nu =0}$. Fit itaque ${\textstyle \mu +\nu }$ par, adeoque 103 residuum quadraticum numeri ${\textstyle 379}$.