Pagina:Werkecarlf02gausrich.djvu/135

Exemplum secundum
${\textstyle m=7,p=49,h=1+2}$

$${\begin{array}{c|c||c|c||c|c||c|c||c|c}{\text{Ind.}}&{\text{Residuum}}&{\text{Ind.}}&{\text{Residuum}}&{\text{Ind.}}&{\text{Residuum}}&{\text{Ind.}}&{\text{Residuum}}\\\hline 0&+1{\phantom {\;+0i}}&10&-1-{\phantom {1}}i&20&{\phantom {+0}}+2i&30&+2-2i&40&+3{\phantom {\;+0i}}\\1&+1+2i&11&+1-3i&21&+3+2i&31&&41&+3-{\phantom {1}}i\\2&-3-3i&12&{\phantom {+0}}-{\phantom {1}}i&22&-1+{\phantom {1}}i&32&+2{\phantom {\;+0i}}&42&-2-2i\\[-4pt]3&+3-2i&13&+2-{\phantom {1}}i&23&-3-{\phantom {1}}i&33&{\phantom {+0}}-3i&43&+2+{\phantom {1}}i\\4&{\phantom {+0}}-3i&14&-3+3i&24&-1{\phantom {\;+0i}}&34&+1+{\phantom {1}}i&44&{\phantom {+0}}-2i\\5&-1-3i&15&-2-3i&25&-1-2i&35&-1+3i&45&-3-2i\\[-4pt]6&-2+2i&16&-3{\phantom {\;+0i}}&26&+3{\phantom {\;+0i}}&36&&46&+1-{\phantom {1}}i\\7&+1-2i&17&-3+{\phantom {1}}i&27&-3+2i&37&-2+{\phantom {1}}i&47&+3+{\phantom {1}}i\\8&-2{\phantom {\;+0i}}&18&+2+2i&28&{\phantom {+0}}+3i&38&+3-3i&&\\9&-2+3i&19&-2-{\phantom {1}}i&29&+1+3i&39&+2+3i&&\end{array}}$$

55.

Adiicimus circa radices primitivas et algorithmum indicum quasdam observationes, demonstrationibus propter facilitatem omissis.

I. Indices secundum modulum ${\textstyle p-1}$ congrui in systemate dato residuis secundum modulum ${\textstyle m}$ congruis respondent et vice versa.

II. Residua, quae respondent indicibus ad ${\textstyle p-1}$ primis, etiam sunt radices primitivae et vice versa.

III. Si accepta radice primitiva ${\textstyle h}$ pro basi, radicis alius primitivae ${\textstyle h^{\prime }}$ index est ${\textstyle t}$, et vice versa ${\textstyle t^{\prime }}$ index ipsius ${\textstyle h}$, dum ${\textstyle h^{\prime }}$ pro basi accipitur, erit ${\textstyle tt^{\prime }\equiv 1{\pmod {p-1}}}$; et si iisdem positis indices cuiusdam alius numeri in his duobus systematibus resp. sunt ${\textstyle u}$, ${\textstyle u^{\prime }}$, erit ${\textstyle tu^{\prime }\equiv u}$, ${\textstyle t^{\prime }u\equiv u^{\prime }{\pmod {p-1}}}$.

IV. Dum numeri ${\textstyle 1}$,${\textstyle 1+i}$ eorumque terni socii (tamquam nimis ieiuni) a modulis nobis considerandis excluduntur, restant numeri primi ii, quos in art. 34 tertio et quarto loco posuimus. Posteriorum normae erunt numeri primi reales formae ${\textstyle 4n+1}$; priorum normae autem quadrata numerorum primorum realium imparium: in utroque igitur casu ${\textstyle p-1}$ per ${\textstyle 4}$ divisibilis est.

V. Denotando indicem numeri ${\textstyle -1}$ per ${\textstyle u}$, erit ${\textstyle 2u\equiv 0{\pmod {p-1}}}$, adeoque vel ${\textstyle u\equiv 0}$, vel ${\textstyle u\equiv {\frac {1}{2}}(p-1)}$: at quum index ${\textstyle 0}$ respondeat residuo ${\textstyle +1}$, index numeri ${\textstyle -1}$ necessario debet esse ${\textstyle {\frac {1}{2}}(p-1)}$.

VI. Perinde denotando per ${\textstyle u}$ indicem numeri ${\textstyle i}$, erit ${\textstyle 2u\equiv {\frac {1}{2}}(p-1){\pmod {p-1}}}$, adeoque vel ${\textstyle u\equiv {\frac {1}{4}}(p-1)}$ vel ${\textstyle u\equiv {\frac {3}{4}}(p-1)}$. Sed hic ambiguitas ab electione radicis primitivae pendet. Scilicet si radice primitiva ${\textstyle h}$ pro basi ac-