Pagina:Werkecarlf02gausrich.djvu/72

$${\begin{aligned}&a=\beta b+c\\&b=\gamma c+d\\&c=\delta d+e\\&d=\varepsilon e+f{\text{ etc.}}\end{aligned}}$$

Hoc modo in serie numerorum continuo decrescentium ${\textstyle b}$, ${\textstyle c}$, ${\textstyle d}$, ${\textstyle e}$, ${\textstyle f}$ etc. tandem ad unitatem perveniemus, quum per hyp. ${\textstyle a}$ et ${\textstyle b}$ sint inter se primi, ita ut aequatio ultima fiat

$$k=\lambda l+1$$

Quum manifesto habeatur

$${\begin{aligned}&{\left[{\frac {a}{b}}\right]=\left[\beta +{\frac {c}{b}}\right]=\beta +\left[{\frac {c}{b}}\right]}\\&{\left[{\frac {2a}{b}}\right]=\left[2\beta +{\frac {2c}{b}}\right]=2\beta +\left[{\frac {2c}{b}}\right]}\\&{\left[{\frac {3a}{b}}\right]=\left[3\beta +{\frac {3c}{b}}\right]=3\beta +\left[{\frac {3c}{b}}\right]}\end{aligned}}$$

etc., erit

$$\varphi (b,a)=\varphi (b,c)+{\frac {1}{2}}\beta (b^{\prime }b^{\prime }+b^{\prime })$$

et proin

$$\varphi (a,b)=a^{\prime }b^{\prime }-{\frac {1}{2}}\beta (b^{\prime }b^{\prime }+b^{\prime })-\varphi (b,c)$$

Per similia ratiocinia fit, si statuimus ${\textstyle \left[{\frac {1}{2}}c\right]=c^{\prime }}$, ${\textstyle \left[{\frac {1}{2}}d\right]=d^{\prime }}$, ${\textstyle \left[{\frac {1}{2}}e\right]=e^{\prime }}$ etc.,

$${\begin{aligned}&\varphi (b,c)=b^{\prime }c^{\prime }-{\frac {1}{2}}\gamma (c^{\prime }c^{\prime }+c^{\prime })-\varphi (a,d)\\&\varphi (c,d)=c^{\prime }d^{\prime }-{\frac {1}{2}}\delta (d^{\prime }d^{\prime }+d^{\prime })-\varphi (d,e)\\&\varphi (d,e)=d^{\prime }e^{\prime }-{\frac {1}{2}}\varepsilon (e^{\prime }e^{\prime }+e^{\prime })-\varphi (e,f)\end{aligned}}$$

etc. usque ad

$$\varphi (k,l)=k^{\prime }l^{\prime }-{\frac {1}{2}}\lambda (l^{\prime }l^{\prime }+l^{\prime })-\varphi (l,1)$$

Hinc, quoniam manifesto est ${\textstyle \varphi (l,1)=0}$, colligimus formulam

$${\begin{aligned}&\varphi (a,b)=a^{\prime }b^{\prime }-b^{\prime }c^{\prime }+c^{\prime }d^{\prime }-d^{\prime }e^{\prime }+{\text{etc.}}\pm k^{\prime }l^{\prime }\\&\quad -{\frac {1}{2}}\beta (b^{\prime }b^{\prime }+b^{\prime })+{\frac {1}{2}}\gamma (c^{\prime }c^{\prime }+c^{\prime })-{\frac {1}{2}}\delta (d^{\prime }d^{\prime }+d^{\prime })+{\frac {1}{2}}\varepsilon (e^{\prime }e^{\prime }+e^{\prime })-{\text{etc.}}\mp {\frac {1}{2}}\lambda (l^{\prime }l^{\prime }+l^{\prime })\end{aligned}}$$

4.

Facile iam ex iis, quae in demonstratione tertia exposita sunt, colligitur, relationem numeri ${\textstyle b}$ ad ${\textstyle a}$, quoties ${\textstyle a}$ sit numerus primus, sponte cognosci e va-