Pagina:Werkecarlf02gausrich.djvu/64

similique ratione etiam erit

$$\beta +\gamma ={\frac {1}{4}}(m-1)(M-1)$$

Iunctis omnibus numeris classium II, IV, VI, manifesto habebimus omnes numeros infra ${\textstyle {\frac {1}{2}}mM}$, qui alicui residuo ex ${\textstyle F^{\prime }}$ secundum modulum ${\textstyle M}$ congrui sunt. Iidem vero numeri ita quoque exhiberi possunt:

$$F^{\prime },M+F^{\prime },2M+F^{\prime },3M+F^{\prime }\ldots {\frac {1}{2}}(m-3)M+F^{\prime }$$

unde omnium multitudo erit ${\textstyle ={\frac {1}{4}}(m-1)(M-1)}$, sive habebimus

$$\beta +\delta +N={\frac {1}{4}}(m-1)(M-1)$$

Perinde e iunctione omnium classium III, IV, VIII colligere licet

$$\gamma +\delta +n={\frac {1}{4}}(m-1)(M-1)$$

Ex his quatuor aequationibus oriuntur sequentes:

$${\begin{aligned}&2\alpha ={\frac {1}{4}}(m-1)(M-1)+n+N\\&2\beta ={\frac {1}{4}}(m-1)(M-1)+n-N\\&2\gamma ={\frac {1}{4}}(m-1)(M-1)-n+N\\&2\delta ={\frac {1}{4}}(m-1)(M-1)-n-N\end{aligned}}$$

quarum quaelibet theorematis veritatem monstrat.

2.

Quodsi iam supponimus, ${\textstyle m}$ et ${\textstyle M}$ esse numeros primos, e combinatione theorematis praecedentis cum lemmate art. 1 theorema fundamentale protinus demanabit. Patet enim,

I. quoties uterque ${\textstyle m}$, ${\textstyle M}$, sive alteruter tantum, sit formae ${\textstyle 4k+1}$, numerum ${\textstyle {\frac {1}{4}}(m-1)(M-1)}$ fore parem, adeoque ${\textstyle n}$ et ${\textstyle N}$ vel simul pares vel simul impares, et proin vel utrumque ${\textstyle m}$ et ${\textstyle M}$ alterius residuum quadraticum, vel utrumque alterius non-residuum quadraticum.

II. Quoties autem uterque ${\textstyle m}$, ${\textstyle M}$ est formae ${\textstyle 4k+3}$, erit ${\textstyle {\frac {1}{4}}(m-1)(M-1)}$ impar, hinc unus numerorum ${\textstyle n}$, ${\textstyle N}$ par, alter impar, et proin unus numerorum ${\textstyle m}$, ${\textstyle M}$ alterius residuum quadraticum, alter alterius non-residuum quadraticum. Q.E.D.