Pagina:Werkecarlf02gausrich.djvu/52

ipsius ${\textstyle W}$ per methodum primam ita perficitur. Pendeant numeri ${\textstyle P}$, ${\textstyle P^{\prime }}$, ${\textstyle P^{\prime \prime }}$ etc., ${\textstyle Q}$, ${\textstyle Q^{\prime }}$, ${\textstyle Q^{\prime \prime }}$ etc. ita a relationibus numerorum ${\textstyle {\frac {n}{p}}}$, ${\textstyle {\frac {n}{p^{\prime }}}}$, ${\textstyle {\frac {n}{p^{\prime \prime }}}}$ etc., ${\textstyle {\frac {n}{q}}}$, ${\textstyle {\frac {n}{q^{\prime }}}}$, ${\textstyle {\frac {n}{q^{\prime \prime }}}}$ etc. ad numeros ${\textstyle p}$, ${\textstyle p^{\prime }}$, ${\textstyle p^{\prime \prime }}$ etc., ${\textstyle q}$, ${\textstyle q^{\prime }}$, ${\textstyle q^{\prime \prime }}$ etc. resp., ut statuatur

$${\begin{aligned}&P=+1,{\text{ si }}{\frac {n}{p}}{\text{ est residuum quadraticum ipsius }}p\\&P=-1,{\text{ si }}{\frac {n}{p}}{\text{ est non-residuum quadraticum ipsius }}p\end{aligned}}$$

et perinde de reliquis. Tunc erit ${\textstyle W}$ productum e factoribus ${\textstyle P\surd p}$, ${\textstyle P^{\prime }\surd p^{\prime }}$, ${\textstyle P^{\prime \prime }\surd p^{\prime \prime }}$ etc., ${\textstyle iQ\surd q}$, ${\textstyle iQ^{\prime }\surd q^{\prime }}$, ${\textstyle iQ^{\prime \prime }\surd q^{\prime \prime }}$ etc., adeoque

$$W=PP^{\prime }P^{\prime \prime }\ldots QQ^{\prime }Q^{\prime \prime }\ldots i^{m}\surd n$$

Per methodum secundam, aut potius statim per praecepta art. 19, erit

${\textstyle W=+\surd n}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +1}$, vel quod eodem redit, si ${\textstyle m}$ est par

${\textstyle W=+i{\sqrt {n}}}$, si ${\textstyle n}$ est formae ${\textstyle 4\mu +3}$, vel si ${\textstyle m}$ est impar

Utrumque casum simul complecti licet per formulam sequentem:

$$W=i^{mm}\surd n$$

Hinc itaque colligitur

$$PP^{\prime }P^{\prime \prime }\ldots QQ^{\prime }Q^{\prime \prime }\ldots =i^{mm-m}$$

Sed ${\textstyle i^{mm-m}}$ fit ${\textstyle =1}$, quoties ${\textstyle m}$ est formae ${\textstyle 4\mu }$ vel ${\textstyle 4\mu +1}$, atque ${\textstyle =-1}$, quoties ${\textstyle m}$ est formae ${\textstyle 4\mu +2}$ vel ${\textstyle 4\mu +3}$, unde deducimus sequens elegantissimum

Theorema. Denotantibus ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. numeros primos impares positivos inaequales, quorum productum statuitur ${\textstyle =n}$, et inter quos ${\textstyle m}$ sint formae ${\textstyle 4\mu +3}$, reliqui formae ${\textstyle 4\mu +1}$: multitudo eorum ex his numeris ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc., quorum non-residua resp. sunt ${\textstyle {\frac {n}{a}}}$, ${\textstyle {\frac {n}{b}}}$, ${\textstyle {\frac {n}{c}}}$ etc., par erit, quoties ${\textstyle m}$ est formae ${\textstyle 4\mu }$ vel ${\textstyle 4\mu +1}$, impar vero, quoties ${\textstyle m}$ est formae ${\textstyle 4\mu +2}$ vel ${\textstyle 4\mu +3}$.

Ita e.g. statuendo ${\textstyle a=3}$, ${\textstyle b=5}$, ${\textstyle c=7}$, ${\textstyle d=11}$, habemus tres numeros formae ${\textstyle 4\mu +3}$, puta ${\textstyle 3}$, ${\textstyle 7}$ et ${\textstyle 11}$; est autem ${\textstyle 5.7.11R3}$; ${\textstyle 3.7.11R5}$; ${\textstyle 3.5.11R7}$; ${\textstyle 3.5.7N11}$, sive unicus ${\textstyle {\frac {n}{d}}}$ est non-residuum ipsius ${\textstyle d}$.

33.

Celeberrimum theorema fundamentale circa residua quadratica nihil aliud est, nisi casus specialis theorematis modo evoluti. Limitando scilicet multitudinem