Pagina:Werkecarlf02gausrich.djvu/89

$${\begin{array}{c|c|c|c}p=37&p=41&p=53&p=61\\2,1,2,4&0,4,3,2&2,3,6,2&4,3,2,6\\2,2,4,1&4,2,2,2&4,4,2,3&3,3,6,3\\2,2,2,2&3,2,3,2&2,4,2,4&4,3,4,3\\2,4,1,2&2,2,2,4&4,2,3,4&3,6,3,3\\\hline p=73&p=89&p=97\\5,6,4,2&3,8,6,4&2,6,7,8\\6,2,5,5&8,4,5,5&6,8,5,5\\4,5,4,5&6,5,6,5&7,5,7,5\\2,5,5,6&4,5,5,8&8,5,5,6\\\end{array}}$$

Quum moduli formae ${\textstyle 8n+1}$ et ${\textstyle 8n+5}$ diverso modo se habeant, utrosque seorsim tractare oportet: a prioribus initium faciemus.

16.

Character ${\textstyle (00)}$ indicat, quot modis diversis aequationi ${\textstyle \alpha +1=\alpha ^{\prime }}$ satisfieri possit, denotantibus ${\textstyle \alpha }$, ${\textstyle \alpha ^{\prime }}$ indefinite numeros e complexu ${\textstyle A}$. Quum pro modulo formae ${\textstyle 8n+1}$, qualem hic subintelligimus, ${\textstyle \alpha ^{\prime }}$ et ${\textstyle p-\alpha ^{\prime }}$ ad eundem complexum pertineant, concinnius dicemus, ${\textstyle (00)}$ exprimere multitudinem modorum diversorum, aequationi ${\textstyle 1+\alpha +\alpha ^{\prime }=p}$, satisfaciendi: manifesto huius aequationis vice etiam congruentia ${\textstyle 1+\alpha +\alpha ^{\prime }\equiv 0{\pmod {p}}}$ fungi potest.

Perinde

${\textstyle (01)}$ indicat multitudinem solutionum congruentiae ${\textstyle 1+\alpha +\beta \equiv 0{\pmod {p}}}$

${\textstyle (02)}$ multitudinem solutionum congruentiae ${\textstyle 1+\alpha +\gamma \equiv 0}$

${\textstyle (03)}$ multitudinem solutionum congruentiae ${\textstyle 1+\alpha +\delta \equiv 0}$

${\textstyle (11)}$ multitudinem solutionum congruentiae ${\textstyle 1+\beta +\beta ^{\prime }\equiv 0}$ etc.

exprimendo indefinite per ${\textstyle \beta }$ et ${\textstyle \beta ^{\prime }}$ numeros e complexu ${\textstyle B}$, per ${\textstyle \gamma }$ numeros e complexu ${\textstyle C}$, per ${\textstyle \delta }$ numeros e complexu ${\textstyle D}$. Hinc statim colligimus sex aequationes sequentes:

$$(01)=(10),\;(02)=(20),\;(03)=(30),\;(12)=(21),\;(13)=(31),\;(23)=(32)$$

E quavis solutione data congruentiae ${\textstyle 1+\alpha +\beta \equiv 0}$ demanat solutio congruentiae ${\textstyle 1+\delta +\delta ^{\prime }\equiv 0}$, accipiendo pro ${\textstyle \delta }$ numerum inter limites ${\textstyle 1\ldots p-1}$