Pagina:Werkecarlf02gausrich.djvu/142

Quum character numeri compositi aequalis sit (sive secundum modulum 4 congruus) aggregato characterum singulorum factorum, sufficit, si pro modulo dato characteres numerorum primorum assignare possumus. Porro quum characteres unitatum ${\textstyle -1}$, ${\textstyle i}$, ${\textstyle -i}$ manifesto sint congrui numeris ${\textstyle {\frac {1}{2}}(p-1)}$, ${\textstyle {\frac {1}{4}}(p-1)}$, ${\textstyle {\frac {3}{4}}(p-1)}$ secundum modulum 4, etiam sufficiet, characteres numerorum inter associatos primariorum exhibuisse. Denique quam moduli secundum modulum ${\textstyle m}$ congrui eundem characterem habeant, sufficit, characteres talium numerorum in tabulam recipere, qui continentur in systemate residuorum absolute minimorum. Praeterea per ratiocinium simile ut in art. 58 demonstratur, si pro modulo ${\textstyle a+bi}$ character numeri ${\textstyle A+Bi}$ sit ${\textstyle \lambda }$, pro modulo ${\textstyle a-bi}$ autem ${\textstyle \lambda ^{\prime }}$ sit character numeri ${\textstyle A-Bi}$, semper esse ${\textstyle \lambda \equiv -\lambda ^{\prime }{\pmod {4}}}$, sive ${\textstyle \lambda +\lambda ^{\prime }}$ per 4 divisibilem: quapropter sufficit, in tabulam recipere modulos, in quibus ${\textstyle b}$ est vel ${\textstyle 0}$ vel positivus.

Ita e.g. si quaeritur character numeri ${\textstyle 11-6i}$ respectu moduli ${\textstyle -5-6i}$, substituimus loco horum numerorum hosce ${\textstyle 11+6i}$, ${\textstyle -5+6i}$; dein determinamus (art. 43) residuum absolute minimum numeri ${\textstyle 11+6i}$ secundum modulum ${\textstyle -5+6i}$, quod fit ${\textstyle -1-4i=-1\times (1+4i)}$; quare quum pro modulo ${\textstyle -5+6i}$ character ipsius ${\textstyle -1}$ sit ${\textstyle 30}$, character numeri ${\textstyle 1+4i}$ autem, ex tabula, ${\textstyle 2}$, erit ${\textstyle 32}$ sive ${\textstyle 0}$ character numeri ${\textstyle 11+6i}$ pro modulo ${\textstyle -5+6i}$, et proin per observationem ultimam etiam character numeri ${\textstyle 11-6i}$ pro modulo ${\textstyle -5-6i}$. Perinde si quaeritur character numeri ${\textstyle -5+6i}$ respectu moduli ${\textstyle 11+6i}$, illius residuum absolute minimum ${\textstyle 1-5i}$ resolvitur in factores ${\textstyle -i}$, ${\textstyle 1+i}$, ${\textstyle 3-2i}$, quibus respondent characteres ${\textstyle 117}$, ${\textstyle 0}$, ${\textstyle 1}$, unde character quaesitus erit ${\textstyle 118}$ sive ${\textstyle 2}$; idem character etiam numero ${\textstyle -5-6i}$ respectu moduli ${\textstyle 11-6i}$ tribuendus est.

$${\begin{array}{l|c|l}{\text{Modulus.}}&{\text{Character.}}&{\text{Numeri.}}\\\hline -3&3&{\phantom {+}}1+i\\+3+2i&3&{\phantom {+}}1+i\\+1+4i&1&-1+2i\\&3&{\phantom {+}}1+i\\-5+2i&0&-1-2i\\&1&{\phantom {+}}1+i\\&2&-1+2i\\-1+6i&0&-3\\&1&1+i,-1+2i\end{array}}$$