Pagina:Werkecarlf02gausrich.djvu/132

51.

Quae in Sectione tertia Disquisitionum Arithmeticarum circa residua potestatum tradita sunt, ad maximam partem, levibus mutationibus adhibitis, etiam in arithmetica numerorum complexorum valent: quinadeo demonstrationes theorematum plerumque retineri possent. Ne tamen quid desit, theoremata principalia demonstrationibus concisis firmata proferemus, ubi semper subintelligendum est, modulum esse numerum primum.

Theorema. Denotante ${\textstyle k}$ integrum per modulum ${\textstyle m}$, cuius norma ${\textstyle =p}$, non divisibilem, erit ${\textstyle k^{p-1}\equiv 1{\pmod {m}}}$.

Demonstr. Constituant ${\textstyle a}$, ${\textstyle b}$, ${\textstyle c}$ etc. systema completum residuorum incongruorum pro modulo ${\textstyle m}$, ita tamen, ut residuum per ${\textstyle m}$ divisibile omissum sit, adeoque multitudo illorum numerorum, quorum complexum denotamus per ${\textstyle C}$, sit ${\textstyle =p-1}$. Sit porro ${\textstyle C^{\prime }}$ complexus productorum ${\textstyle ka}$, ${\textstyle kb}$, ${\textstyle kc}$ etc. Ex his productis per hyp. nullum erit divisibile per ${\textstyle m}$, quare singula habebunt residua congrua in complexu ${\textstyle C}$, puta fieri poterit ${\textstyle ak\equiv a^{\prime }}$, ${\textstyle bk\equiv b^{\prime }}$, ${\textstyle ck\equiv c^{\prime }}$ etc. ${\textstyle {\pmod {m}}}$, ita ut numeri ${\textstyle a^{\prime }}$, ${\textstyle b^{\prime }}$, ${\textstyle c^{\prime }}$ etc. ipsi in complexu ${\textstyle C}$ inveniantur: denotemus complexum numerorum ${\textstyle a^{\prime }}$, ${\textstyle b^{\prime }}$, ${\textstyle c^{\prime }}$ etc. per ${\textstyle C^{\prime \prime }}$. Sint ${\textstyle P}$, ${\textstyle P^{\prime }}$, ${\textstyle P^{\prime \prime }}$ producta e singulis numeris complexuum ${\textstyle C}$, ${\textstyle C^{\prime }}$, ${\textstyle C^{\prime \prime }}$ resp., sive

$${\begin{aligned}&P=abc\ldots \\&P^{\prime }=k^{p-1}abc\ldots =k^{p-1}P\\&P^{\prime \prime }=a^{\prime }b^{\prime }c^{\prime }\ldots \end{aligned}}$$

Quum numeri complexus ${\textstyle C^{\prime \prime }}$ deinceps congrui sint numeris complexus ${\textstyle C^{\prime }}$, erit ${\textstyle P^{\prime \prime }\equiv P^{\prime }}$ sive ${\textstyle P^{\prime \prime }\equiv k^{p-1}P}$. At quum facile perspiciatur, binos quosvis numeros complexus ${\textstyle C^{\prime \prime }}$ inter se incongruos, adeoque omnes inter se diversos esse, necessario numeri complexus ${\textstyle C^{\prime \prime }}$ cum numeris complexus ${\textstyle C}$ prorsus conveniunt, ordine tantummodo mutato, unde fit ${\textstyle P^{\prime \prime }=P}$. Erit itaque ${\textstyle (k^{p-1}-1)P}$ numerus per ${\textstyle m}$ divisibilis, unde, quum ${\textstyle m}$ sit numerus primus singulos factores ipsius ${\textstyle P}$ non metiens, necessario ${\textstyle k^{p-1}-1}$ per ${\textstyle m}$ divisibilis esse debebit. Q. E. D.

52.

Theorema. Denotante ${\textstyle k}$, ut in art. praec., integrum per modulum ${\textstyle m}$ non divisibilem, atque ${\textstyle t}$ exponentem minimum (praeter 0${\textstyle )}$, pro quo ${\textstyle k^{t}\equiv 1{\pmod {m}}}$, erit ${\textstyle t}$ divisor cuiusvis alius exponentis ${\textstyle u}$, pro quo ${\textstyle k^{u}\equiv 1{\pmod {m}}}$.