Pagina:Werkecarlf02gausrich.djvu/85

$${\begin{aligned}&{\begin{array}{c}p=89\\g=3,f=34\end{array}}\\{\begin{array}{l}A\\\;\\B\\\;\\C\\\;\\D\\\;\end{array}}&{\begin{array}{|rrrrrrrrrrrrrrrrrrrrrr}1,&2,&4,&8,&11,&16,&22,&25,&32,&39,&44,&45,&50,&57,&64,&67,&73,&78,&81,&85,\\&&87,&88\\3,&6,&7,&12,&14,&23,&24,&28,&33,&41,&43,&46,&48,&56,&61,&65,&66,&75,&77,&82\\&&83,&86\\5,&9,&10,&17,&18,&20,&21,&34,&36,&40,&42,&47,&49,&53,&55,&68,&69,&71,&72,&79\\&&80,&84&\\13,&15,&19,&26,&27,&29,&30,&31,&35,&37,&38,&51,&52,&54,&58,&59,&60,&62,&63,&70\\&&74,&76\end{array}}\\&{\begin{array}{c}p=97\\g=5,f=22\end{array}}\\{\begin{array}{l}A\\\;\\B\\\;\\C\\\;\\D\\\;\end{array}}&{\begin{array}{|rrrrrrrrrrrrrrrrrrrrrr}1,&4,&6,&9,&16,&22,&24,&33,&35,&36,&43,&47,&50,&54,&61,&62,&64,&73,&75,&81,\\&&88,&91,&93,&96\\5,&13,&14,&17,&19,&20,&21,&23,&29,&30,&41,&45,&52,&56,&67,&68,&74,&76,&77,&78\\&&80,&83,&84,&92\\2,&3,&8,&11,&12,&18,&25,&27,&31,&32,&44,&48,&49,&53,&65,&66,&70,&72,&79,&85\\&&86,&89,&94,&95\\7,&10,&15,&26,&28,&34,&37,&38,&39,&40,&42,&46,&51,&55,&57,&58,&59,&60,&63,&69\\&&71,&82,&87,&90\end{array}}\end{aligned}}$$

12.

Quum numerus 2 sit residuum quadraticum omnium numerorum primorum formae ${\textstyle 8n+1}$, non-residuum vero omnium formae ${\textstyle 8n+5}$, pro modulis primis formae prioris 2 in classe ${\textstyle A}$ vel ${\textstyle C}$, pro modulis formae posterioris in classe ${\textstyle B}$ vel ${\textstyle D}$ invenietur. Quum discrimen inter classes ${\textstyle B}$ et ${\textstyle D}$ non sit essentiale, quippe quod tantummodo ab electione numeri ${\textstyle f}$ pendet, modulos formae ${\textstyle 8n+5}$ aliquantisper seponemus. Modulos formae ${\textstyle 8n+1}$ autem inductioni subiiciendo, invenimus 2 pertinere ad ${\textstyle A}$ pro ${\textstyle p=73}$, 89, 113, 233, 257, 281, 337, 353 etc.; contra 2 pertinere ad ${\textstyle C}$ pro ${\textstyle p=17}$, 41, 97, 137, 193, 241, 313, 401, 409, 433, 449, 457 etc.

Ceterum quum pro modulo primo formae ${\textstyle 8n+1}$ numerus ${\textstyle -1}$ sit residuum biquadraticum, patet, ${\textstyle -2}$ semper cum ${\textstyle +2}$ ad eandem classem referendum esse.