Pagina:Werkecarlf02gausrich.djvu/108

$${\begin{array}{l}A,{\text{ quoties }}b\equiv 0{\pmod {5}}\\B,{\text{ quoties }}b\equiv a\\C,{\text{ quoties }}a\equiv 0\\D,{\text{ quoties }}b\equiv 4a\end{array}}$$

Manifestum est, has regulas complecti casus omnes, quum pro ${\textstyle b\equiv 2a}$, vel ${\textstyle b\equiv 3a{\pmod {5}}}$, fieret ${\textstyle aa+bb\equiv 0}$, Q.E.A., quum per hypothesin ${\textstyle p}$ sit numerus primus a 5 diversus.

27.

Perinde inductio ad numeros ${\textstyle -7}$, ${\textstyle -11}$, ${\textstyle +13}$, ${\textstyle +17}$, ${\textstyle -19}$, ${\textstyle -23}$ applicata satisque producta sequentes regulas indigitat:

Pro numero ${\textstyle -7}$

$${\begin{array}{l|l}A&a\equiv 0,{\text{ vel }}b\equiv 0{\pmod {7}}\\B&b\equiv 4a,{\text{ vel }}b\equiv 5a\\C&b\equiv a,{\text{ vel }}b\equiv 6a\\D&b\equiv 2a,{\text{ vel }}b\equiv 3a\end{array}}$$

Pro numero ${\textstyle -11}$

$${\begin{array}{l|l}A&b\equiv 0,5a,{\text{ vel }}6a{\pmod {11}}\\B&b\equiv a,3a{\text{ vel }}4a\\C&a\equiv 0,{\text{ vel }}b\equiv 2a{\text{ vel }}9a\\D&b\equiv 7a,8a{\text{ vel }}10a\end{array}}$$

Pro numero ${\textstyle +13}$

$${\begin{array}{l|l}A&b\equiv 0,4a,9a{\pmod {13}}\\B&b\equiv 6a,11a,12a\\C&a\equiv 0;b\equiv 3a,10a\\D&b\equiv a,2a,7a\end{array}}$$

Pro numero ${\textstyle +17}$

$${\begin{array}{l|l}A&a\equiv 0;b\equiv 0,a,16a{\pmod {17}}\\B&b\equiv 2a,6a,8a,14a\\C&b\equiv 5a,7a,10a,12a\\D&b\equiv 3a,9a,11a,15a\end{array}}$$