Pagina:Werkecarlf02gausrich.djvu/134

$$C^{{\frac {p-1}{c^{\gamma }}}\cdot {\frac {p-1}{a}}}\equiv 1,{\text{ et sic porro; quapropter esse debet }}A^{{\frac {p-1}{a^{\alpha }}}\cdot {\frac {p-1}{a}}}\equiv 1$$ Iam determinetur integer positivus ${\textstyle \lambda }$ talis, ut fiat $$\lambda b^{\beta }c^{\gamma }\ldots \equiv 1{\pmod {a}}$$ quod fieri poterit, quum numerus primus ${\textstyle a}$ ipsum ${\textstyle b^{\beta }c^{\gamma }\ldots }$ non metiatur, statuaturque ${\textstyle \lambda b^{\beta }c^{\gamma }\ldots =1+a\mu }$. Manifesto fit $$A^{\lambda \cdot {\frac {p-1}{a^{\alpha }}}\cdot {\frac {p-1}{a}}}\equiv 1,{\text{ sive, quoniam }}\lambda \cdot {\frac {p-1}{a^{\alpha }}}\cdot {\frac {p-1}{a}}=(1+a\mu ){\frac {p-1}{a}}=(p-1)\mu +{\frac {p-1}{a}}$$ habemus ${\textstyle A^{(p-1)\mu }\cdot A^{\frac {p-1}{a}}\equiv 1}$, atque hinc, quum sponte sit ${\textstyle A^{(p-1)\mu }\equiv 1}$, etiam ${\textstyle A^{\frac {p-1}{a}}\equiv 1}$, quod est contra hypothesin. Suppositio itaque, ${\textstyle t}$ esse submultiplum ipsius ${\textstyle p-1}$, consistere nequit, eritque adeo necessario ${\textstyle h}$ radix primitiva.

54.

Denotante ${\textstyle h}$ radicem primitivam pro modulo ${\textstyle m}$, cuius norma ${\textstyle =p}$, termini progressionis $$1,h,hh,h^{3}\ldots h^{p-2}$$ inter se incongrui erunt, unde facile colligitur, quemlibet integrum non divisibilem per modulum uni ex istis congruum esse debere, sive illam seriem exhibere systema completum residuorum incongruorum exclusa cifra. Exponens eius potestatis, cui numerus datus congruus est, vocari potest huius index, dum ${\textstyle h}$ tamquam basis consideratur. Ecce quaedam exempla, ubi cuivis indici residuum absolute minimum apposuimus.

Exemplum Primum
${\textstyle m=5+4i,\quad p=41,\quad h=1+2i}$

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