Pagina:Werkecarlf02gausrich.djvu/66

tate diminutae divisibiles erunt per ${\textstyle 1-x^{p}}$; quamobrem in hoc casu etiam ${\textstyle f(x^{n})-p}$ per ${\textstyle 1-x^{p}}$ et proin etiam per ${\textstyle {\frac {1-x^{p}}{1-x}}}$ divisibilis erit.

3.

Theorema. Statuendo

$$x-x^{\alpha }+x^{\alpha \alpha }-x^{\alpha ^{3}}+x^{\alpha ^{4}}-{\text{etc.}}-x^{\alpha ^{p-2}}=\xi$$

erit ${\textstyle \xi \xi \mp p}$ divisibilis per ${\textstyle {\frac {1-x^{p}}{1-x}}}$, accepto signo superiori, quoties ${\textstyle p}$ est formae ${\textstyle 4k+1}$, inferiori, quoties ${\textstyle p}$ est formae ${\textstyle 4k+3}$.

Demonstr. Facile perspicietur, ex ${\textstyle p-1}$ functionibus hisce

$${\begin{aligned}&+x\xi -xx+x^{\alpha +1}-x^{\alpha \alpha +1}+{\text{etc.}}+x^{\alpha ^{p-2}+1}\\&-x^{\alpha }\xi -x^{2\alpha }+x^{\alpha \alpha +\alpha }-x^{\alpha ^{3}+\alpha }+{\text{etc.}}+x^{\alpha ^{p-1}+\alpha }\\&+x^{\alpha \alpha }\xi -x^{2\alpha \alpha }+x^{\alpha ^{3}+\alpha \alpha }-x^{\alpha ^{4}+\alpha \alpha }+{\text{etc.}}+x^{\alpha ^{p}+\alpha \alpha }\\&-x^{\alpha ^{3}}\xi -x^{2\alpha ^{3}}+x^{\alpha ^{4}+\alpha ^{3}}-x^{\alpha ^{5}+\alpha ^{3}}+{\text{etc.}}+x^{\alpha ^{p+1}+\alpha ^{3}}\end{aligned}}$$

etc. usque ad

$$-x^{\alpha ^{p-2}}\xi -x^{2\alpha ^{p-2}}+x^{\alpha ^{p-1}+\alpha ^{p-2}}-x^{\alpha ^{p}+\alpha ^{p-2}}+{\text{etc.}}+x^{\alpha ^{2p-4}+\alpha ^{p-2}}$$

primam fieri ${\textstyle =0}$, singulas reliquas autem per ${\textstyle 1-x^{p}}$ divisibiles. Quare per ${\textstyle 1-x^{p}}$ etiam divisibilis erit omnium summa, quae colligitur

$${\begin{aligned}=\xi \xi &-(f(xx)-1)+(f(x^{\alpha +1})-1)-(f(x^{\alpha \alpha +1})-1)\\&+(f(x^{\alpha ^{3}+1})-1)-{\text{etc.}}+(f(x^{\alpha ^{p-2}+1})-1)\\=\xi \xi &-f(xx)+f(x^{\alpha +1})-f(x^{\alpha \alpha +1})+f(x^{\alpha ^{3}+1})-{\text{etc.}}+f(x^{\alpha ^{p-2}+1})=\Omega \end{aligned}}$$

Erit itaque haecce expressio ${\textstyle \Omega }$ etiam divisibilis per ${\textstyle {\frac {1-x^{p}}{1-x}}}$. Iam inter exponentes ${\textstyle 2}$, ${\textstyle \alpha +1}$, ${\textstyle \alpha \alpha +1}$, ${\textstyle \alpha ^{3}+1\ldots \alpha ^{p-2}+1}$ unicus tantum erit divisibilis per ${\textstyle p}$, puta ${\textstyle \alpha ^{{\frac {1}{2}}(p-1)}+1}$, unde per art. praec. singulae partes expressionis ${\textstyle \Omega }$ hae

$$f(xx),f(x^{\alpha +1}),f(x^{\alpha \alpha +1}),f(x^{\alpha ^{3}+1}){\text{ etc.}}$$

excepto solo termino ${\textstyle f(x^{\alpha ^{{\frac {1}{2}}(p-1)}+1})}$, divisibiles erunt per ${\textstyle {\frac {1-x^{p}}{1-x}}}$. Istas itaque partes delere licebit, ita ut per ${\textstyle {\frac {1-x^{p}}{1-x}}}$ etiam divisibilis maneat functio

$$\xi \xi \mp f(x^{\alpha ^{{\frac {1}{2}}(p-1)}+1})$$