Pagina:Werkecarlf02gausrich.djvu/87

ties ipsorum multitudo impar sit, fieri formae ${\textstyle 8m+3}$ vel ${\textstyle 8m+5}$, idemque adeo in hoc casu valere de producto ${\textstyle a}$.

Ex his omnibus itaque colligitur theorema elegans:

Quoties ${\textstyle a}$ est formae ${\textstyle 8m+1}$ vel ${\textstyle 8m+7}$, numerus ${\textstyle 2}$ in complexu ${\textstyle A}$ contentus erit; quoties vero ${\textstyle a}$ est formae ${\textstyle 8m+3}$ vel ${\textstyle 8m+5}$, numerus ${\textstyle 2}$ in complexu ${\textstyle C}$ invenietur.

Quod confirmatur per exempla in art. praec. enumerata; priores enim moduli ita discerpuntur: ${\textstyle 73=1+2.36}$, ${\textstyle 89=81+2.4}$, ${\textstyle 113=81+2.16}$, ${\textstyle 233=225+2.4}$, ${\textstyle 257=225+2.16}$, ${\textstyle 281=81+2.100}$, ${\textstyle 337=49+2.144}$, ${\textstyle 353=225+2.64}$; posteriores vero ita: ${\textstyle 17=9+2.4}$, ${\textstyle 41=9+2.16}$, ${\textstyle 97=25+2.36}$, ${\textstyle 137=9+2.64}$, ${\textstyle 193=121+2.36}$, ${\textstyle 241=169+2.36}$, ${\textstyle 313=25+2.144}$, ${\textstyle 401=9+2.196}$, ${\textstyle 409=121+2.144}$, ${\textstyle 433=361+2.36}$, ${\textstyle 449=441+2.4}$, ${\textstyle 457=169+2.144}$.

14.

Quum discerptio numeri ${\textstyle p}$ in quadratum simplex et duplex nexum tam insignem cum classificatione numeri ${\textstyle 2}$ prodiderit, operae pretium esse videtur tentare, num discerptio in duo quadrata, cui numerum ${\textstyle p}$ aeque obnoxium esse constat, similem forte successum suppeditet. Ecce itaque discerptiones numerorum ${\textstyle p}$, pro quibus ${\textstyle 2}$ pertinet ad classem

$${\begin{array}{c|c}A&C\\\hline {\begin{aligned}9&+64\\25&+64\\49&+64\\169&+64\\1&+256\\25&+256\\81&+256\\289&+64\\\\\\\\\\\end{aligned}}&{\begin{aligned}1&+16\\25&+16\\81&+16\\121&+16\\49&+144\\225&+16\\169&+144\\1&+400\\9&+400\\289&+144\\49&+400\\441&+16\end{aligned}}\end{array}}$$