Pagina:Werkecarlf02gausrich.djvu/18

Ceterum simili modo demonstrari potest, si ${\textstyle k}$ fuerit numerus par ad ${\textstyle p}$ primus, fore

$$\left.{\begin{array}{l}{\phantom {+}}\left[{\frac {k}{p}}\right]+\left[{\frac {2k}{p}}\right]+\left[{\frac {3k}{p}}\right]\ldots +\left[{\frac {{\frac {1}{2}}(p-1)k}{p}}\right]\\+\left[{\frac {p}{k}}\right]+\left[{\frac {2p}{k}}\right]+\left[{\frac {3p}{k}}\right]\ldots +\left[{\frac {{\frac {1}{2}}kp}{k}}\right]\end{array}}\right\}={\frac {1}{4}}k(p-1)$$

At huic propositioni ad institutum nostrum non necessariae non immoramur.

7.

Iam ex combinatione theorematis praec. cum propos. VIII. art. 4. theorema fundamentale protinus demanat. Nimirum denotantibus ${\textstyle k}$, ${\textstyle p}$ numeros primos positivos inaequales quoscunque, et ponendo

$${\begin{aligned}(k,p)+\left[{\frac {k}{p}}\right]+\left[{\frac {2k}{p}}\right]+\left[{\frac {3k}{p}}\right]\ldots +\left[{\frac {{\frac {1}{2}}(p-1)k}{p}}\right]&=L\\(p,k)+\left[{\frac {p}{k}}\right]+\left[{\frac {2p}{k}}\right]+\left[{\frac {3p}{k}}\right]\ldots +\left[{\frac {{\frac {1}{2}}(k-1)p}{k}}\right]&=M\end{aligned}}$$

per VIII. art. 4. patet, ${\textstyle L}$ et ${\textstyle M}$ semper fieri numeros pares. At per theorema art. 6. erit

$$L+M=(k,p)+(p,k)+{\frac {1}{4}}(k-1)(p-1)$$

Quoties igitur ${\textstyle {\frac {1}{4}}(k-1)(p-1)}$ par evadit, quod fit, si vel uterque ${\textstyle k}$, ${\textstyle p}$ vel saltem alteruter est formae ${\textstyle 4n+1}$, necessario ${\textstyle (k,p)}$ et ${\textstyle (p,k)}$ vel ambo pares vel ambo impares esse debent. Quoties autem ${\textstyle {\frac {1}{4}}(k-1)(p-1)}$ impar est, quod evenit, si uterque ${\textstyle k}$, ${\textstyle p}$ est formae ${\textstyle 4n+3}$, necessario alter numerorum ${\textstyle (k,p)}$, ${\textstyle (p,k)}$ par, alter impar esse debebit. In casu priori itaque relatio ipsius ${\textstyle k}$ ad ${\textstyle p}$ et relatio ipsius ${\textstyle p}$ ad ${\textstyle k}$ (quatenus alter alterius residuum vel non-residuum est) identicae erunt, in casu posteriori oppositae. Q. E. D.