Pagina:Werkecarlf02gausrich.djvu/126

${\textstyle ay-bx=\eta }$, erit ${\textstyle \eta \equiv k\xi }$, ${\textstyle \xi \equiv -k\eta {\pmod {p}}}$. Omnes itaque numeri ${\textstyle \xi +\eta i}$, quibus residua simpliciter minima ${\textstyle x+yi}$ respondent, habebuntur, dum vel pro ${\textstyle \xi }$ deinceps accipiuntur valores ${\textstyle 0}$, ${\textstyle 1}$, ${\textstyle 2}$, ${\textstyle 3\ldots p-1}$, et pro ${\textstyle \eta }$ residua minima positiva productorum ${\textstyle k\xi }$ secundum modulum ${\textstyle p}$, vel ordine alio pro ${\textstyle \eta }$ illi valores et pro ${\textstyle \xi }$ residua minima productorum ${\textstyle -k\eta }$. E singulis ${\textstyle \xi +\eta i}$ dein respondentes ${\textstyle x+yi}$ invenientur per formulam

$$x+yi={\frac {\xi +\eta i}{a-bi}}={\frac {a\xi -b\eta }{p}}+{\frac {a\eta +b\xi }{p}}\cdot i$$

Ceterum obvium est, ${\textstyle \eta }$, dum ${\textstyle \xi }$ unitate crescat, vel augmentum ${\textstyle k}$ vel decrementum ${\textstyle p-k}$ pati, adeoque ${\textstyle x+yi}$

vel mutationem ${\textstyle {\frac {a-kb}{p}}+{\frac {ak+b}{p}}\cdot i}$ vel hanc ${\textstyle {\frac {a-kb}{p}}+b+({\frac {ak+b}{p}}-a)i}$

quae observatio ad constructionem faciliorem reddendam inservit.

Denique patet, si residua absolute minima ${\textstyle x+yi}$ desiderentur, haec praecepta eatenus tantum mutari, quatenus ipsi ${\textstyle \xi }$ deinceps tribuendi sint valores inter limites ${\textstyle -{\frac {1}{2}}p}$ et ${\textstyle +{\frac {1}{2}}p}$, dum pro ${\textstyle \eta }$ accipere oporteat residua absolute minima productorum ${\textstyle k\xi }$. Ecce conspectum residuorum minimorum pro modulo ${\textstyle 5+2i}$ hoc modo adornatorum:

Residua simpliciter minima.

$${\begin{array}{r|r||r|r||r|r}\xi +\eta i&x+yi&\xi +\eta i&x+yi&\xi +\eta i&x+yi\\\hline 0&0&10+25i&+5i&20+21i&+2+5i\\1+17i&-1+3i&11+13i&+1+3i&21+9i&+3+3i\\2+5i&+i&12+i&+2+i&22+26i&+2+6i\\3+22i&+1+4i&13+18i&+1+4i&23+14i&+3+4i\\4+10i&+2i&14+6i&+2+2i&24+2i&+4+2i\\5+27i&-1+5i&15+23i&+1+5i&25+19i&+3+5i\\6+15i&+3i&16+11i&+2+3i&26+7i&+4+3i\\7+3i&+1+i&17+28i&+1+6i&27+24i&+3+6i\\8+20i&+4i&18+16i&+2+4i&28+12i&+4+4i\\9+8i&+1+2i&19+4i&+3+2i&&\\\end{array}}$$