Pagina:Werkecarlf02gausrich.djvu/139

In theoria residuorum quadraticorum character numeri ${\textstyle \alpha +\beta i}$ respectu moduli ${\textstyle a+bi}$ idem est, qui numeri ${\textstyle \alpha -\beta i}$ respectu moduli ${\textstyle a-bi}$.

Demonstratio huius theorematis inde petitur, quod uterque modulus eandem normam ${\textstyle p}$ habet, atque quoties ${\textstyle (\alpha +\beta i)^{{\frac {1}{2}}(p-1)}-1}$ per ${\textstyle a+bi}$ divisibilis est, etiam ${\textstyle (\alpha -\beta i)^{{\frac {1}{2}}(p-1)}-1}$ per ${\textstyle a-bi}$ divisibilis evadit, quoties autem ${\textstyle (\alpha +\beta i)^{{\frac {1}{2}}(p-1)}+1}$ per ${\textstyle a+bi}$ divisibilis est, etiam ${\textstyle (\alpha -\beta i)^{{\frac {1}{2}}(p-1)}+1}$ per ${\textstyle a-bi}$ divisibilis esse debet.

59.

Progrediamur ad numeros primos impares.

Numerum ${\textstyle -1+2i}$ invenimus esse residuum quadraticum modulorum ${\textstyle +3+2i}$, ${\textstyle +1-4i}$, ${\textstyle -5+2i}$, ${\textstyle -5-2i}$, ${\textstyle -1-6i}$, ${\textstyle +7-2i}$, ${\textstyle -3+8i}$, ${\textstyle +5+8i}$, ${\textstyle +5-8i}$, ${\textstyle +9+4i}$ etc.

non-residuum autem modulorum ${\textstyle -1-2i}$, ${\textstyle -3}$, ${\textstyle +3-2i}$, ${\textstyle +1+4i}$, ${\textstyle -1+6i}$, ${\textstyle +5+4i}$, ${\textstyle +5-4i}$, ${\textstyle -7}$, ${\textstyle +7+2i}$, ${\textstyle -5+6i}$, ${\textstyle -5-6i}$, ${\textstyle -3-8i}$, ${\textstyle +9-4i}$ etc.

Reducendo modulos prioris classis ad residua eorum absolute minima secundum modulum ${\textstyle -1+2i}$, haec sola invenimus ${\textstyle +1}$ et ${\textstyle -1}$, puta ${\textstyle +3+2i\equiv -1}$, ${\textstyle +1-4i\equiv -1}$, ${\textstyle -5+2i\equiv +1}$, ${\textstyle -5-2i\equiv -1}$ etc.

Contra omnes moduli posterioris classis congrui inveniuntur secundum modulum ${\textstyle -1+2i}$ vel ipsi ${\textstyle +i}$, vel ipsi ${\textstyle -i}$.

At numeri ${\textstyle +1}$, ${\textstyle -1}$ ipsi sunt residua quadratica moduli ${\textstyle -1+2i}$, atque ${\textstyle +i}$ et ${\textstyle -i}$ eiusdem non-residua: quocirca, quatenus inductioni fidem habere licet, prodit theorema: Numerus ${\textstyle -1+2i}$ est residuum vel non-residuum quadraticum numeri primi ${\textstyle a+bi}$, prout hic est residuum vel non-residuum quadraticum ipsius ${\textstyle -1+2i}$, siquidem ${\textstyle a+bi}$ est primarius e quaternis associatis, vel potius, si ${\textstyle a}$ est impar, ${\textstyle b}$ par.

Ceterum ex hoc theoremate sponte sequuntur theoremata analoga circa numeros ${\textstyle +1-2i}$, ${\textstyle -1-2i}$, ${\textstyle +1+2i}$.

60.

Instituendo similem inductionem circa numerum ${\textstyle -3}$ vel ${\textstyle +3}$, invenimus, utrumque esse residuum quadraticum modulorum ${\textstyle +3+2i}$, ${\textstyle +3-2i}$,