Pagina:Werkecarlf02gausrich.djvu/158

Formula itaque pro ${\textstyle g}$ in artt. 73 et 74 inventa, transit in sequentem

$$g={\frac {1}{8}}((a+b)^{2}-1)+2n-4N$$

statuendo ${\textstyle a\mp 1=4n}$, ubi ${\textstyle n}$ erit integer. Sed quum hinc habeatur ${\textstyle 1=16nn-8an+aa}$, formula haec etiam sequenti modo exhiberi potest:

$$g={\frac {1}{8}}(-aa+2ab+bb+1)+4({\frac {1}{2}}(a+1)n-nn-N)$$

Quapropter quum ${\textstyle g}$ sit character numeri ${\textstyle -1-i}$ pro modulo ${\textstyle a+bi}$, hic character fit ${\textstyle \equiv {\frac {1}{8}}(-aa+2ab+bb+1){\pmod {4}}}$, quod est ipsum theorema supra (art.64) per inductionem erutum, sponteque inde demanant theoremata circa characteres numerorum ${\textstyle 1+i}$, ${\textstyle 1-i}$, ${\textstyle -1+i}$. Quamobrem haec quatuor theoremata, pro casu eo, ubi ${\textstyle a}$ et ${\textstyle b}$ sunt positivi, iam rigorose sunt demonstrata.

76.

Si manente ${\textstyle a}$ positivo ${\textstyle b}$ est negativus, statuatur ${\textstyle b=-b^{\prime }}$, ut fiat ${\textstyle b^{\prime }}$ positivus. Quum iam evictum sit, ita pro modulo ${\textstyle a+b^{\prime }i}$ characterem numeri ${\textstyle -1-i}$ esse ${\textstyle \equiv {\frac {1}{8}}(-aa+2ab^{\prime }+b^{\prime }b^{\prime }+1){\pmod {4}}}$, character numeri ${\textstyle -1+i}$ pro modulo ${\textstyle a-b^{\prime }i}$ per theorema in art. 62 prolatum erit ${\textstyle \equiv {\frac {1}{8}}(aa-2ab^{\prime }-b^{\prime }b^{\prime }-1)}$, i.e. character numeri ${\textstyle -1+i}$ pro modulo ${\textstyle a+bi}$ fit ${\textstyle \equiv {\frac {1}{8}}(aa+2ab-bb-1)}$: hoc vero est ipsum theorema in art. 64 allatum, unde tria reliqua circa characteres numerorum ${\textstyle 1+i}$, ${\textstyle 1-i}$, ${\textstyle -1-i}$ sponte demanant. Quapropter ista theoremata etiam pro casu, ubi ${\textstyle b}$ negativus est, demonstrata sunt, scilicet pro omnibus casibus, ubi ${\textstyle a}$ est positivus.

Denique si ${\textstyle a}$ est negativus, statuatur ${\textstyle a=-a^{\prime },b=-b^{\prime }}$. Quum itaque per iam demonstrata character numeri ${\textstyle 1+i}$ respectu moduli ${\textstyle a^{\prime }+b^{\prime }i}$ sit ${\textstyle \equiv {\frac {1}{8}}(-a^{\prime }a^{\prime }+2a^{\prime }b^{\prime }-3b^{\prime }b^{\prime }+1){\pmod {4}}}$, nihilque intersit, utrum numerum ${\textstyle a^{\prime }+b^{\prime }i}$ an oppositum ${\textstyle -a^{\prime }-b^{\prime }i}$ moduli loco habeamus; manifesto character numeri ${\textstyle 1+i}$ respectu moduli ${\textstyle a+bi}$ est ${\textstyle \equiv {\frac {1}{8}}(-aa+2ab-3bb+1)}$, et similia valent circa characteres numerorum ${\textstyle 1-i}$, ${\textstyle -1+i}$, ${\textstyle -1-i}$.

Ex his itaque colligitur, demonstrationem theorematum circa characteres numerorum ${\textstyle 1+i}$, ${\textstyle 1-i}$, ${\textstyle -1+i}$, ${\textstyle -1-i}$ (artt. 63. 64) nulli amplius limitationi obnoxiam esse.