Pagina:Werkecarlf02gausrich.djvu/128

res, divisores communes maximi appellandi erunt, horumque norma erit multiplum normae cuiusvis alius divisoris communis.

Si resolutio duorum numerorum propositorum in factores simplices non adest, divisor communis maximus adiumento similis algorithmi eruitur, ut pro numeris realibus. Sint ${\textstyle m}$, ${\textstyle {m}^{\prime }}$ duo numeri propositi, formeturque per divisionem repetitam series ${\textstyle m^{\prime \prime }}$, ${\textstyle m^{\prime \prime \prime }}$ etc. ita, ut ${\textstyle m^{\prime \prime }}$ sit residuum absolute minimum ipsius ${\textstyle m}$ secundum modulum ${\textstyle m^{\prime }}$, dein ${\textstyle m^{\prime \prime }}$ residuum absolute minimum ipsius ${\textstyle m^{\prime }}$ secundum modulum ${\textstyle m^{\prime \prime }}$ et sic porro. Denotando normas numerorum ${\textstyle m}$, ${\textstyle m^{\prime }}$, ${\textstyle m^{\prime \prime }}$, ${\textstyle m^{\prime \prime \prime }}$ etc. resp. per ${\textstyle p}$, ${\textstyle p^{\prime }}$, ${\textstyle p^{\prime \prime }}$, ${\textstyle p^{\prime \prime \prime }}$ etc., erit ${\textstyle {\frac {p^{\prime \prime }}{p^{\prime }}}}$ norma quotientis ${\textstyle {\frac {m^{\prime \prime }}{m^{\prime }}}}$, adeoque per definitionem residui absolute minimi certo non maior quam ${\textstyle {\frac {1}{2}}}$; idem valet de ${\textstyle {\frac {p^{\prime \prime \prime }}{p^{\prime \prime }}}}$ etc. Quapropter integri reales positivi ${\textstyle p^{\prime }}$, ${\textstyle p^{\prime \prime }}$, ${\textstyle p^{\prime \prime \prime }}$ etc. seriem continuo decrescentem formabunt, unde necessario tandem ad terminum ${\textstyle 0}$ pervenietur, sive, quod idem est, in serie ${\textstyle m}$, ${\textstyle m^{\prime }}$, ${\textstyle m^{\prime \prime }}$, ${\textstyle m^{\prime \prime }}$ etc. tandem ad terminum perveniemus, qui praecedentem absque residuo metitur. Sit hic ${\textstyle m^{(n+1)}}$, statuamusque

$${\begin{aligned}&m=km^{\prime }+m^{\prime \prime }\\&m^{\prime }=k^{\prime }m^{\prime \prime }+m^{\prime \prime \prime }\\&m^{\prime \prime }=k^{\prime \prime }m^{\prime \prime \prime }+m^{\prime \prime \prime \prime }\end{aligned}}$$

etc. usque ad

$$m^{(n)}=k^{(n)}m^{(n+1)}$$

Percurrendo has aequationes ordine inverso, elucet, ${\textstyle m^{(n+1)}}$ singulos terminos praecedentes ${\textstyle m^{(n)}\ldots m^{\prime \prime }}$, ${\textstyle m^{\prime }}$, ${\textstyle m}$ metiri; percurrendo autem easdem aequationes ordine directo, manifestum est, quemvis divisorem communem numerorum ${\textstyle m}$, ${\textstyle m}$ etiam metiri singulos sequentes. Conclusio prior docet, ${\textstyle m^{(n+1)}}$ esse divisorem communem numerorum ${\textstyle m}$, ${\textstyle m^{\prime }}$; posterior autem, hunc divisorem esse maximum.

Ceterum quoties residuum ultimum ${\textstyle m^{(n+1)}}$ alicui quatuor unitatum ${\textstyle 1}$, ${\textstyle -1}$, ${\textstyle i}$, ${\textstyle -i}$ aequale evadit, hoc indicium erit, ${\textstyle m}$ et ${\textstyle m^{\prime }}$ inter se primos esse.

47.

Si aequationes art. praec., omissa ultima, ita combinantur, ut ${\textstyle m^{\prime \prime },m^{\prime \prime \prime }}$, ${\textstyle m^{\prime \prime \prime \prime }\ldots m^{(n)}}$ eliminentur, orietur aequatio talis

$$m^{(n+1)}=hm+h^{\prime }m^{\prime }$$