Pagina:Werkecarlf02gausrich.djvu/154

Statuendo ${\textstyle ay-bx=\eta }$, ${\textstyle (a+b)x-(a-b)y=\zeta }$, ${\textstyle p-2ax-2by=\theta }$, criterium numerorum ${\textstyle x+yi}$ ad complexum ${\textstyle G}$ pertinentium consistit in tribus conditionibus, ut ${\textstyle \eta }$, ${\textstyle \zeta }$, ${\textstyle \theta }$ sint numeri positivi. Quum fiat ${\textstyle px=(a-b)\eta +a\zeta }$, ${\textstyle p(a-2x)=a\theta +2b\eta }$, manifestum est, ${\textstyle x}$ et ${\textstyle 2a-x}$ esse debere numeros positivos, sive ${\textstyle x}$ alicui numerorum ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots {\frac {1}{2}}(a-1)}$ aequalem. Porro quum sit ${\textstyle (a-b)\theta =2b\zeta +p(a-b-2x)}$, patet, quamdiu ${\textstyle x}$ minor sit quam ${\textstyle {\frac {1}{2}}(a-b)}$, conditionem secundam (iuxta quam ${\textstyle \zeta }$ positivus esse debet) iam implicare tertiam (quod ${\textstyle \theta }$ debet esse positivus); contra quoties ${\textstyle x}$ sit maior quam ${\textstyle {\frac {1}{2}}(a-b)}$, conditionem secundam iam contineri sub tertia. Quamobrem pro valoribus ipsius ${\textstyle x}$ his ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots {\frac {1}{2}}(a-b-1)}$ tantummodo prospiciendum est, ut ${\textstyle \eta }$ et ${\textstyle \zeta }$ positivi evadant, sive ut ${\textstyle y}$ maior sit quam ${\textstyle {\frac {bx}{a}}}$ et minor quam ${\textstyle {\frac {(a+b)x}{a-b}}}$: pro valore itaque tali dato ipsius ${\textstyle x}$ aderunt numeri ${\textstyle x+yi}$ omnino

$$\left[{\frac {(a+b)x}{a-b}}\right]-\left[{\frac {bx}{a}}\right]$$

si uncis in eadem significatione utimur, qua iam alibi passim usi sumus (Conf. Theorematis arithm. dem. nova art. 4 et Theorematis fund. in doctr. de residuis quadr. etc. Algorithm. nov. art. 3). Contra pro valoribus ipsius ${\textstyle x}$ his ${\textstyle {\frac {1}{2}}(a-b+1)}$, ${\textstyle {\frac {1}{2}}(a-b+3)\ldots {\frac {1}{2}}(a-1)}$ sufficiet, ut ipsis ${\textstyle \eta }$ et ${\textstyle \theta }$ valores positivi concilientur, sive ut ${\textstyle y}$ maior sit quam ${\textstyle {\frac {bx}{a}}}$ et minor quam ${\textstyle {\frac {p-2ax}{2b}}}$ sive ${\textstyle {\frac {1}{2}}b+{\frac {aa-2ax}{2b}}}$: quare pro valore tali dato ipsius ${\textstyle x}$ aderunt numeri ${\textstyle x+yi}$ omnino

$$\left[{\frac {1}{2}}b+{\frac {aa-2ax}{2b}}\right]-\left[{\frac {bx}{a}}\right]$$

Hinc itaque colligimus, multitudinem numerorum complexus ${\textstyle G}$ esse

$$g=\Sigma \left[{\frac {(a+b)x}{a-b}}\right]+\Sigma \left[{\frac {1}{2}}b+{\frac {aa-2ax}{2b}}\right]-\Sigma \left[{\frac {bx}{a}}\right]$$

ubi in termino primo summatio extendenda est per omnes valores integros ipsius ${\textstyle x}$ ab ${\textstyle 1}$ usque ad ${\textstyle {\frac {1}{2}}(a-b-1)}$, in secundo ab ${\textstyle {\frac {1}{2}}(a-b+1)}$ usque ad ${\textstyle {\frac {1}{2}}(a-1)}$, in tertio ab ${\textstyle 1}$ usque ad ${\textstyle {\frac {1}{2}}(a-1)}$.

Si characteristica ${\textstyle \varphi }$ in eadem significatione utimur, ut loco citato (Theorematis fund. etc. Algor. nov. art. 3), puta ut sit

$$\varphi (t,u)=\left[{\frac {u}{t}}\right]+\left[{\frac {2u}{t}}\right]+\left[{\frac {3u}{t}}\right]\ldots +\left[{\frac {t^{\prime }u}{t}}\right]$$

denotantibus ${\textstyle t}$, ${\textstyle u}$ numeros positivos quoscunque, atque ${\textstyle t^{\prime }}$ numerum ${\textstyle \left[{\frac {1}{2}}t\right]}$, terminus ille primus fit