Pagina:Gauss, Carl Friedrich - Werke (1870).djvu/132

122

de formis secundi gradus.

$bb-ac\equiv (\mu (mb+nc)-\nu (ma+nb))^{2}{\pmod {M}}$ i. e. $bb-ac$ residuum quadraticum ipsius $M$.

Numerum $bb-ac$, a cuius indole proprietates formae $(a,b,c)$ imprimis pendere, in sequentibus docebimus, determinantem huius formae vocabimus.

Valores expr. $\textstyle \surd {(bb-ac)}{\pmod {M}}$, ad quos repraesentatio numeri $M$ per formam $(a,b,c)$ pertinet.

155.

Erit itaque $\mu (mb+nc)-\nu (ma+nb)$ valor expressionis $\textstyle \surd {(bb-ac)}{\pmod {M}}$ Constat autem, numeros $\mu$, $\nu$ infinitis modis ita determinari posse ut sit $\mu m+\nu n=1$, unde alii aliique valores illius expressionis prodibunt, qui quem nexum inter se habeant videamus. Sit non modo $\mu m+\nu n=1$, sed etiam $\mu 'm+\nu 'n=1$, ponaturque $\mu (mb+nc)-\nu (ma+nb)=v$, $\mu '(mb+nc)-\nu '(ma+nb)=v'$ Multiplicando aequationem $\mu m+\nu n=1$ per $\mu '\!$, alteram $\mu 'm+\nu 'n=1$ per $\mu$, et subtrahendo fit $\mu '-\mu =n(\mu '\nu -\mu \nu ')$ similiterque multiplicando illam per $\nu '\!$, hanc per $\nu$, fit subtrahendo $\nu '-\nu =m(\mu \nu '-\mu '\nu )$. Hinc statim prodit $v'-v=(\mu '\nu -\mu \nu ')(amm+2bmn+cnn)=(\mu '\nu -\mu \nu ')M$ sive $\textstyle v'\equiv v{\pmod {M}}$. Quomodocunque igitur $\mu$, $\nu$ determinentur, formula $\mu (mb+nc)-\nu (ma+nb)$ valores diversos (i. e. incongruos) expressionis $\textstyle \surd {(bb-ac)}{\pmod {M}}$ dare nequit. Si itaque $v$ est valor quicunque illius formulae: repraesentationem numeri $M$ per formam $axx+2bxy+cyy$ eam ubi $x=m$, $y=n$, pertinere dicemus ad valorem $v$ expressionis $\textstyle \surd {(bb-ac)}{\pmod {M}}$. Ceterum facile ostendi potest, si valor formulae illius aliquis sit $v$ atque $\textstyle v'\equiv v{\pmod {M}}$, loco numerorum $\mu$, $\nu$, qui dant $v$, alios $\mu '\!$, $\nu '\!$ accipi posse, qui dent $v'\!$. Scilicet faciendo $\mu '=\mu +{\tfrac {n(v'-v)}{M}}$, $\nu '=\nu +{\tfrac {m(v'-v)}{M}}$ fiet $\mu 'm+\nu 'n=\mu m+\nu n=1$