Pagina:Werkecarlf02gausrich.djvu/138

58.

Quum diiudicatio characteris numeri compositi, utrum sit residuum quadraticum an non-residuum, pendeat a characteribus factorum, manifesto sufficiet, si evolutionem criteriorum ad distinguendos modulos, pro quibus numerus datus ${\textstyle k}$ sit residuum quadraticum, ab iis, pro quibus sit non-residuum, ad tales valores ipsius ${\textstyle k}$ limitemus, qui sint numeri primi, insuperque inter associatos primarii. In qua investigatione inductio protinus theoremata maxime elegantia suppeditat.

Incipiamus a numero ${\textstyle 1+i}$, qui invenitur esse residuum quadraticum modulorum
${\textstyle -1+2i}$, ${\textstyle +3-2i}$, ${\textstyle -5-2i}$, ${\textstyle -1-6i}$, ${\textstyle +5+4i}$, ${\textstyle +5-4i}$, ${\textstyle -7}$, ${\textstyle +7+2i}$, ${\textstyle -5+6i}$, etc.
non-residuum quadraticum autem sequentium
${\textstyle -1-2i}$, ${\textstyle -3}$, ${\textstyle +3+2i}$, ${\textstyle +1+4i}$, ${\textstyle +1-4i}$, ${\textstyle -5+2i}$, ${\textstyle -1+6i}$, ${\textstyle +7-2i}$, ${\textstyle -5-6i}$, ${\textstyle -3+8i}$, ${\textstyle -3-8i}$, ${\textstyle +5+8i}$, ${\textstyle +5-8i}$, ${\textstyle +9+4i}$, ${\textstyle +9-4i}$ etc.

Si hunc conspectum, in quo semper e quaternis modulis associatis primarium apposuimus, attente examinamus, facile animadvertimus, modulos ${\textstyle a+bi}$ in priori classe omnes esse tales, pro quibus ${\textstyle a+b}$ fiat ${\textstyle \equiv +1{\pmod {8}}}$, in posteriori vero tales, pro quibus ${\textstyle a+b\equiv -3{\pmod {8}}}$. Manifesto hoc criterium, si loco moduli primarii ${\textstyle m}$ adoptamus associatum ${\textstyle -m}$, ita immutari debet, ut pro modulis prioris classis sit ${\textstyle a+b\equiv -1}$, pro modulis posterioris ${\textstyle \equiv +3{\pmod {8}}}$. Quare, siquidem inductio non fefellerit, generaliter, designante ${\textstyle a+bi}$ numerum primum, in quo ${\textstyle a}$ impar, ${\textstyle b}$ par, ${\textstyle 1+i}$ fit eius residuum quadraticum vel non-residuum quadraticum, prout ${\textstyle a+b\equiv \pm 1}$, vel ${\textstyle \equiv \pm 3{\pmod {8}}}$.

Pro numero ${\textstyle -1-i}$ eadem regula valet, quae pro ${\textstyle 1+i}$. Contra considerando ${\textstyle 1-i}$ tamquam productum ex ${\textstyle -i}$ in ${\textstyle 1+i}$, manifestum est, numero ${\textstyle 1-i}$ eundem characterem competere, qui tribuendus sit ipsi ${\textstyle 1+i}$, quoties ${\textstyle b}$ sit pariter par, oppositum autem, quoties ${\textstyle b}$ sit impariter par, unde facile colligitur, ${\textstyle 1-i}$ esse residuum quadraticum numeri primi ${\textstyle a+bi}$, quoties sit ${\textstyle a-b\equiv \pm 1}$, nonresiduum autem, quoties habeatur ${\textstyle a-b\equiv \pm 3{\pmod {8}}}$, semper supponendo, ${\textstyle a}$ esse imparem, ${\textstyle b}$ parem.

Ceterum haec secunda propositio e priori etiam deduci potest adiumento theorematis generalioris, quod ita enunciamus: