Pagina:Werkecarlf02gausrich.djvu/111

${\textstyle +3}$ pertinet

${\textstyle {\begin{array}{c|l}{\text{ad complexum}}&{\text{si}}\\A&b\equiv 0{\pmod {12}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 2{\pmod {4}}\\B&b\equiv 8a{\text{ vel }}10a{\pmod {12}}\\C&b\equiv 6a{\pmod {12}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 0{\pmod {4}}\\D&b\equiv 2a{\text{ vel }}4a{\pmod {12}}\end{array}}}$

Perinde criteria pro ${\textstyle \pm 6}$ petuntur e combinatione criteriorum pro ${\textstyle \mp 2}$ et ${\textstyle -3}$; scilicet

${\textstyle +6}$ pertinet

${\textstyle {\begin{array}{c|l}{\text{ad complexum}}&{\text{si}}\\A&b\equiv 0,{\phantom {00a}}2a,22a{\pmod {24}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 4a{\pmod {8}}\\B&b\equiv 4a,{\phantom {00}}6a,{\phantom {0}}8a{\pmod {24}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 2a{\pmod {8}}\\C&b\equiv 10a,12a,14a{\pmod {24}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 0{\phantom {a}}{\pmod {8}}\\D&b\equiv 16a,18a,20a{\pmod {24}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 6a{\pmod {8}}\end{array}}}$

${\textstyle -6}$ vero

${\textstyle {\begin{array}{c|l}{\text{ad complexum}}&{\text{si}}\\A&b\equiv 0,{\phantom {a}}10a,14a{\pmod {24}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 4a{\pmod {8}}\\B&b\equiv 4a,{\phantom {0}}8a,18a{\pmod {24}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 6a{\pmod {8}}\\C&b\equiv 2a,12a,22a{\pmod {24}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 0{\phantom {a}}{\pmod {8}}\\D&b\equiv 6a,16a,20a{\pmod {24}};{\text{ vel simul }}a\equiv 0{\pmod {3}},b\equiv 2a{\pmod {8}}\end{array}}}$
Simili modo criteria pro numero ${\textstyle +21}$ concinnabuntur e criteriis pro ${\textstyle -3}$ et ${\textstyle -7}$; criteria pro ${\textstyle -105}$ e criteriis pro ${\textstyle -1}$, ${\textstyle -3}$, ${\textstyle +5}$, ${\textstyle -7}$, etc.

30.

Amplissimam itaque messem theorematum specialium aperit inductio, theoremati pro numero 2 affinium: sed desideratur vinculum commune, desiderantur demonstrationes rigorosae, quum methodus, per quam in commentatione prima numerum 2 absolvimus, ulteriorem applicationem non patiatur. Non desunt quidem methodi diversae, per quas demonstrationibus pro casibus particularibus potiri liceret, iis potissimum, qui distributionem residuorum quadraticorum inter complexus ${\textstyle A}$, ${\textstyle C}$ spectant, quibus tamen non immoramur; quum theoria genera-