Pagina:Werkecarlf02gausrich.djvu/86

13.

Si exempla art. praec. inter se comparantur, primo saltem aspectu criterium nullum simplex se offerre videtur, per quod modulos priores a posterioribus dignoscere liceret. Nihilominus duo huiusmodi criteria dantur, elegantia et simplicitate perinsignia, ad quorum alterum considerationes sequentes viam sternent.

Modulus ${\textstyle p}$, tamquam numerus primus formae ${\textstyle 8n+1}$, reduci poterit, et quidem unico tantum modo, sub formam ${\textstyle aa+2bb}$ (Disquiss. Arithm. art. 182, II); radices ${\textstyle a}$, ${\textstyle b}$ positive accipi supponemus. Manifesto ${\textstyle a}$ impar erit, ${\textstyle b}$ vero par; statuemus autem ${\textstyle b=2^{\lambda }c}$, ita ut ${\textstyle c}$ sit impar. Iam observamus

I. quum habeatur ${\textstyle p\equiv aa{\pmod {c}}}$ ipsum ${\textstyle p}$ esse residuum quadraticum ipsius ${\textstyle c}$, et proin etiam singulorum factorum primorum, in quos ${\textstyle c}$ resolvitur: vicissim itaque, per theorema fundamentale, singuli hi factores primi erunt residua quadratica ipsius ${\textstyle p}$, et proin etiam illorum productum ${\textstyle c}$ erit residuum quadraticum ipsius ${\textstyle p}$. Quod quum etiam de numero 2 valeat, patet, ${\textstyle b}$ esse residuum quadraticum ipsius ${\textstyle p}$, et proin ${\textstyle bb}$, nec non ${\textstyle -bb}$, residuum biquadraticum.

II. Hinc ${\textstyle -2bb}$ ad eandem classem referri debet, in qua invenitur numerus ${\textstyle 2}$; quare quum ${\textstyle aa\equiv -2bb}$, manifestum est, ${\textstyle 2}$ vel in classe ${\textstyle A}$, vel in classe ${\textstyle C}$ inveniri, prout ${\textstyle a}$ sit vel residuum quadraticum ipsius ${\textstyle p}$, vel non-residuum quadraticum.

III. Iam supponamus, ${\textstyle a}$ in factores suos primos resolutum esse, e quibus ii, qui sunt vel formae ${\textstyle 8m+1}$ vel ${\textstyle 8m+7}$, denotentur per ${\textstyle \alpha }$, ${\textstyle \alpha ^{\prime }}$, ${\textstyle \alpha ^{\prime \prime }}$ etc., ii vero, qui sunt vel formae ${\textstyle 8m+3}$ vel ${\textstyle 8m+5}$, per ${\textstyle \beta }$, ${\textstyle \beta ^{\prime }}$, ${\textstyle \beta ^{\prime \prime }}$ etc.: posteriorum multitudo sit ${\textstyle =\mu }$. Quoniam ${\textstyle p\equiv 2bb{\pmod {a}}}$, erit ${\textstyle p}$ residuum quadraticum eorum factorum primorum ipsius ${\textstyle a}$, quorum residuum quadraticum est ${\textstyle 2}$, i.e. factorum ${\textstyle \alpha }$, ${\textstyle \alpha ^{\prime }}$, ${\textstyle \alpha ^{\prime \prime }}$ etc.; non-residuum quadraticum vero factorum eorum, quorum non-residuum quadraticum est ${\textstyle 2}$, i.e. factorum ${\textstyle \beta }$, ${\textstyle \beta ^{\prime }}$, ${\textstyle \beta ^{\prime \prime }}$ etc. Quocirca, vice versa, per theorema fundamentale, singuli ${\textstyle \alpha ^{\prime }}$, ${\textstyle \alpha ^{\prime }}$, ${\textstyle \alpha ^{\prime \prime }}$ etc. erunt residua quadratica ipsius ${\textstyle p}$, singuli ${\textstyle \beta }$, ${\textstyle \beta ^{\prime }}$, ${\textstyle \beta ^{\prime \prime }}$ etc. autem non-residua quadratica. Ex his itaque concluditur, productum ${\textstyle a}$ fore residuum quadraticum ipsius ${\textstyle p}$, vel non-residuum, prout ${\textstyle \mu }$ par sit vel impar.

IV. Sed facile confirmatur, productum omnium ${\textstyle \alpha }$, ${\textstyle \alpha ^{\prime }}$, ${\textstyle \alpha ^{\prime \prime }}$ etc. fieri formae ${\textstyle 8m+1}$ vel ${\textstyle 8m+7}$, idemque valere de producto omnium ${\textstyle \beta }$, ${\textstyle \beta ^{\prime }}$, ${\textstyle \beta ^{\prime \prime }}$ etc., si horum multitudo fuerit par, ita ut in hoc casu etiam productum ${\textstyle a}$ necessario fieri debeat formae ${\textstyle 8m+1}$ vel ${\textstyle 8m+7}$; contra productum omnium ${\textstyle \beta }$, ${\textstyle \beta ^{\prime }}$, ${\textstyle \beta ^{\prime \prime }}$ etc., quo-