![{\displaystyle W=1+2\sum R^{a}=1+R+R^{4}+R^{9}+{\text{etc.}}+R^{(n-1)^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5751af2114aeebdbcb0a4c70d43628adaa97467d)
adeoque
![{\textstyle W=+\surd n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a79ece2a6c9081b7583710f0009e15b7ebb4907)
, si
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
est formae
![{\textstyle 4\mu +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15634a0e9a313acfea7beedc1ae6000fd2e0a6fe)
, atque
![{\textstyle W=+i\surd n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cfe98a6ba34ec5d45b2df1669df76afa90fe64f)
, si
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
est formae
![{\textstyle 4\mu +3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63ee43be5041253e68e3fedbd60bfd613b89fb02)
.
Contra in casu altero, ubi
est non-residuum ipsius
, erit
![{\displaystyle W=1+2\Sigma R^{b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/228bd5f7bd6bc24772535c96808e3746d0c3208e)
Hinc quum manifesto omnes
![{\textstyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a503f107a7c104e40e484cee9e1f5993d28ffd8)
,
![{\textstyle b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a780b69dfc55238880ef18a134dc65260877e2)
complexum integrum numerorum
![{\textstyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6706df9ed9f240d1a94545fb4e522bda168fe8fd)
,
![{\textstyle 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78ed0cd8140e5a15b6fcce83602df58458e0f3b0)
,
![{\textstyle 3\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eac6903a3d9d59a615131be9d564eee3fdfffb3)
expleant, adeoque sit
![{\displaystyle \Sigma R^{a}+\Sigma R^{b}=R+R^{2}+R^{3}+{\text{etc.}}+R^{n-1}=-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82b975bbae9ecbd6e47624403f89cb2921f451a)
fiet
![{\displaystyle W=-1-2\Sigma R^{a}=-(1+R+R^{4}+R^{9}+{\text{etc.}}+R^{(n-1)^{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa5c5a99583eb580f5cdf37743c7223e8632c4bc)
adeoque
![{\textstyle W=-\surd n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2748c2b7da9b6a10da288190aa64d6529a04aa3)
, si
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
est formae
![{\textstyle 4\mu +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15634a0e9a313acfea7beedc1ae6000fd2e0a6fe)
, atque
![{\textstyle W=-i\surd n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea6bdf91309157d83bc8996aaaf0eaa952a802e)
, si
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
est formae
![{\textstyle 4\mu +3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63ee43be5041253e68e3fedbd60bfd613b89fb02)
.
Hinc itaque colligitur
primo, si
est formae
, atque
residuum quadraticum ipsius
,
![{\displaystyle T=+\surd n,\quad U=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2988847c281edb60d6a49cbd14ce0a4433f359cb)
secundo, si
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
est formae
![{\textstyle 4\mu +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15634a0e9a313acfea7beedc1ae6000fd2e0a6fe)
, atque
![{\textstyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0)
non-residuum ipsius
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
,
![{\displaystyle T=-\surd n,\quad U=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f05c2b9c9455b48add8472c5f80552dcfbf630c)
tertio, si
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
est formae
![{\textstyle 4\mu +3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63ee43be5041253e68e3fedbd60bfd613b89fb02)
, atque
![{\textstyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0)
residuum ipsius
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
,
![{\displaystyle T=0,\quad U=+\surd n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8aed426177a3ffc6333d7f64e154982a797b8dd0)
quarto, si
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
est formae
![{\textstyle 4\mu +3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63ee43be5041253e68e3fedbd60bfd613b89fb02)
, atque
![{\textstyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5595fc0c47452f8fc2aa6e29c3611f047714b0)
non-residuum ipsius
![{\textstyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6)
,
![{\displaystyle T=0,\quad U=-\surd n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ee4a24e835427941381c227f46d763a72b90009)
21.
Sit secundo
quadratum altiorve potestas numeri primi imparis
, statuaturque
, ita ut sit
vel
vel
. Hic ante omnia observare convenit, si
sit integer quicunque per
non divisibilis, fieri