Demonstr. I. Sint
,
integri tales qui faciant
, unde erit
![{\displaystyle i=\alpha b-\beta a+m(\beta +\alpha i)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c00cbbe4b88ddd3d6797d5a342fa1332f39a9695)
Proposito itaque numero integro complexo
![{\textstyle A+Bi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07ad2b51b68e614cf77c5c78ee48071f7920e8b1)
, habebimus
![{\displaystyle A+Bi=A+(\alpha b-\beta a)B+m(\beta B+\alpha Bi)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e26adbeccee5082acf509b492be8e16c62bd2367)
Quare denotando per
![{\textstyle h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13fda070627ca694f85f588a432f8158cc4df1e4)
residuum minimum positivum numeri
![{\textstyle A+(\alpha b-\beta a)B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef4b7eae6c1e4fd2f09cb83f00d1bff119a9d590)
secundum modulum
![{\textstyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b)
, statuendoque
![{\displaystyle A+(\alpha b-\beta a)B=h+kp=h+m(ak-bki)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f6f3494cfef7003268f81431cf46efbdf9a5b0a)
erit
![{\displaystyle A+Bi=h+m(\beta B+ak+(\alpha B-bk)i)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a875128a8a4167f1c37214292d3416bdd94ceb34)
sive
![{\displaystyle A+Bi\equiv h{\pmod {m}}.\quad {\text{ Q. E. P. }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0e901bc0b8f4dfab737c05b13e5947d9f24a03)
II. Quoties eidem numero complexo duo numeri reales
,
secundum modulum
congrui sunt, etiam inter se congrui erunt. Statuamus itaque
, unde fit
![{\displaystyle (h-h^{\prime })(a-bi)=p(c+di)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0486a997f4aeafcfae5b32d8ceae291ef82c1b9)
adeoque
![{\displaystyle (h-h^{\prime })a=pc,\quad (h-h^{\prime })b=-pd}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9d66e2a0743640cb19c5ac00b82a65ab1ecb8a)
nec non, propter
![{\textstyle a\alpha +b\beta =1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26badbce1cfd4d9476612f2177d58aae5225f49f)
,
![{\displaystyle h-h^{\prime }=p(c\alpha -d\beta ),{\text{ i.e. }}h\equiv h^{\prime }{\pmod {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86e67ec2e6d3f3b39a24391720f94b9261b2fbca)
Quapropter
et
, siquidem sunt inaequales, ambo simul in complexu numerorum
,
,
,
contenti esse nequeunt. Q. E. S.
41.
Theorema. Secundum modulum complexum
, cuius norma
, et pro quo
,
non sunt inter se primi, sed divisorem communem maximum
habent (quem positive acceptum supponimus), quilibet numerus complexus congruus est residuo
tali, ut
sit aliquis numerorum
, atque
aliquis horum
,
,
,
, et quidem unico tantum inter omnia
residua, quae tali forma gaudent.