Pagina:Werkecarlf02gausrich.djvu/109

Pro numero -19.

$${\begin{array}{l|l}A&b\equiv 0,2a,5a,14a,17a{\pmod {19}}\\B&b\equiv 3a,7a,11a,13a,18a\\C&a\equiv 0;b\equiv 4a,9a,10a,15a\\D&b\equiv a,6a,8a,12a,16a\\\end{array}}$$

Pro numero -23.

$${\begin{array}{l|l}A&a\equiv 0;b\equiv 0,7a,10a,13a,16a{\pmod {23}}\\B&b\equiv 2a,3a,4a,11a,15a,17a\\C&b\equiv a,5a,9a,14a,18a,22a\\D&b\equiv 6a,8a,12a,19a,20a,21a\end{array}}$$

28.

Theoremata specialia hoc modo per inductionem eruta confirmari inveniuntur, quousque haec continuetur, formamque criteriorum pulcherrimam manifestant. Si vero inter se conferuntur, ut conclusiones generales inde petantur, primo statim aspectu se offerunt observationes sequentes.

Criteria diiudicationis, ad quemnam complexum referendus sit numerus primus ${\textstyle \pm q}$ (sumendo signum superius vel inferius, prout ${\textstyle q}$ est formae ${\textstyle 4n+1}$ vel ${\textstyle 4n+3}$), pendent a formis numerorum ${\textstyle a}$, ${\textstyle b}$ inter se collatorum respectu moduli ${\textstyle q}$. Scilicet

I. quoties ${\textstyle a\equiv 0{\pmod {q}},\pm q}$ pertinet ad complexum determinatum, qui est ${\textstyle A}$ pro ${\textstyle q=7}$, ${\textstyle 17}$, ${\textstyle 23}$, nec non ${\textstyle C}$ pro ${\textstyle q=3}$, ${\textstyle 11}$, ${\textstyle 13}$, ${\textstyle 19}$, unde coniectura oritur, casum priorem generaliter valere, quoties ${\textstyle q}$ sit formae ${\textstyle 8n\pm 1}$, posteriorem vero, quoties ${\textstyle q}$ sit formae ${\textstyle 8n\pm 3}$. Ceterum complexus ${\textstyle B}$ et ${\textstyle D}$ iam absque inductione excluduntur pro valore ipsius ${\textstyle a}$ per ${\textstyle q}$ divisibili, ubi fit ${\textstyle p\equiv bb{\pmod {q}}}$, i.e. ubi ${\textstyle p}$ est residuum quadraticum ipsius ${\textstyle q}$, unde per theorema fundamentale ${\textstyle \pm q}$ esse debet residuum quadraticum ipsius ${\textstyle p}$.

II. Quoties autem ${\textstyle a}$ per ${\textstyle q}$ non est divisibilis, criterium pendet a valore expressionis ${\textstyle {\frac {b}{a}}{\pmod {q}}}$. Admittit quidem haec expressio ${\textstyle q}$ valores diversos, puta ${\textstyle 0}$, ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots q-1}$ sed quoties ${\textstyle q}$ est formae ${\textstyle 4n+1}$, excludendi sunt bini valo-