Pagina:Werkecarlf02gausrich.djvu/106

$${\begin{array}{c|c|c}{\text{Pertinet}}&+2&-2\\\hline {\text{ad complexum}}&{\text{si }}b{\text{, secundum modulum}}&8{\text{, fit congruus ipsi}}\\\hline A&0&0\\B&2a&6a\\C&4a&4a\\D&6a&2a\\\end{array}}$$

Facile intelligitur, theoremata sic enunciata haud amplius pendere a conditione ${\textstyle a\equiv 1{\pmod {4}}}$, sed etiamnum valere, si fuerit ${\textstyle a\equiv 3{\pmod {4}}}$, dummodo conditio altera, ${\textstyle af\equiv b{\pmod {p}}}$, conservetur.

Aeque facile perspicitur, summam horum theorematum eleganter contrahi posse in formulam unicam, puta:

si ${\textstyle a}$ et ${\textstyle b}$ positive accipiuntur, semper fit

$$b^{{\frac {1}{2}}ab}\equiv a^{{\frac {1}{2}}ab}2^{{\frac {1}{4}}(p-1)}{\pmod {p}}$$

25.

Videamus nunc, quatenus inductio classificationem numeri 3 indigitet. Tabula art. 11 ulterius continuata (semper adoptata radice primitiva minima), monstrat, ${\textstyle +3}$ pertinere ad complexum

$${\begin{array}{c||c||c||c}A{\text{ pro}}&B{\text{ pro}}&C{\text{ pro}}&D{\text{ pro}}\\{\begin{array}{r|r|r}p&a&b\\13&-3&+2\\109&-3&+10\\181&+9&+10\\193&-7&-12\\229&-15&+2\\277&+9&+14\\&&\\&&\\&&\\&&\\&&\\&&\\&&\\\end{array}}&{\begin{array}{r|r|r}p&a&b\\17&+1&-4\\29&+5&+2\\53&-7&+2\\89&+5&-8\\101&+1&+10\\113&-7&-8\\137&-11&-4\\197&+1&-14\\233&+13&+8\\257&+1&-16\\269&+13&+10\\281&+5&+16\\293&+17&+2\end{array}}&{\begin{array}{r|r|r}p&a&b\\37&+1&-6\\61&+5&-6\\73&-3&-8\\97&+9&+4\\157&-11&-6\\241&-15&-4\\&&\\&&\\&&\\&&\\&&\\&&\\&&\\\end{array}}&{\begin{array}{r|r|r}p&a&b\\5&+1&+2\\41&+5&-4\\149&-7&+10\\173&+13&+2\\&&\\&&\\&&\\&&\\&&\\&&\\&&\\&&\\&&\\\end{array}}\end{array}}$$