denotando per
coëfficientem medium
![{\displaystyle {\frac {{\frac {1}{2}}(p-1)\cdot {\frac {1}{2}}(p-3)\cdot {\frac {1}{2}}(p-5)\ldots {\frac {1}{2}}(p+3)}{1\cdot 2\cdot 3\ldots {\frac {1}{4}}(p-1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06311f209ebfc3a59577f9be34d44ca8246c128a)
Substituendo itaque pro
![{\textstyle x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781)
deinceps numeros
![{\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 p-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66af1e742cbbb7869422addbe3644d921c66ca5a)
, obtinebimus per lemma art. 19
![{\displaystyle \Sigma (x^{4}+1)^{{\frac {1}{2}}(p-1)}\equiv -2-P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f65b98e2f1b05cc4d328db4fb6c44fed56af088)
At perpendendo ea quae in art. 19 exposuimus, insuperque, quod numeri complexuum
![{\textstyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a118c6ad00742b3f5dccd2f0e74b5e369df6fd31)
,
![{\textstyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de0b47ffc21636dc2df68f6c793177a268f10e9b)
,
![{\textstyle C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b)
,
![{\textstyle D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e5200f518cb5afe304ec42ffdd4f6c63c702f77)
, ad potestatem exponentis
![{\textstyle {\frac {1}{2}}(p-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/572700477f7516e6751c405e64bac470f690b93d)
evecti congrui sunt, secundum modulum
![{\textstyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b)
, numeris
![{\textstyle +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9985827c3397c7c045ca72729071a9534ca5043e)
,
![{\textstyle -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03d0251b380d21d87eb1cd07628008e254050c3)
,
![{\textstyle +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9985827c3397c7c045ca72729071a9534ca5043e)
,
![{\textstyle -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e03d0251b380d21d87eb1cd07628008e254050c3)
resp., facile intelligitur fieri
![{\displaystyle \Sigma (x^{4}+1)^{{\frac {1}{2}}(p-1)}\equiv 4(00)-4(01)+4(02)-4(03)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f51b24cf03ab6e50c75e054dd1651bf8a3df7c12)
adeoque per schemata in fine artt. 18, 20 tradita
![{\displaystyle \Sigma (x^{4}+1)^{{\frac {1}{2}}(p-1)}\equiv -2a-2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b31d56bc52de1cc10323d79f436fc324f2435f)
Comparatio horum duorum valorum suppeditat elegantissimum theorema: scilicet habemus
![{\displaystyle P\equiv 2a{\pmod {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb990e907a912be8a6493ad3f1b1a0299198e28)
Denotando quatuor producta
![{\displaystyle {\begin{aligned}&1.2.3\ldots {\frac {1}{4}}(p-1)\\&{\frac {1}{4}}(p+3).{\frac {1}{4}}(p+7).{\frac {1}{4}}(p+11)\ldots {\frac {1}{2}}(p-1)\\&{\frac {1}{2}}(p+1).{\frac {1}{2}}(p+3).{\frac {1}{2}}(p+5)\ldots {\frac {3}{4}}(p-1)\\&{\frac {1}{4}}(3p+1).{\frac {1}{4}}(3p+5).{\frac {1}{4}}(3p+9)\ldots (p-1)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd25320dc858e902190fbc8ec6402462e89e93cd)
resp. per
![{\textstyle q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a32efd4de244331181b8fa97862015251f65da5f)
,
![{\textstyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dfb06630b52c9e18fcc0a4688da10774206729)
,
![{\textstyle s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45013da7f502d373d039cb1056a9c4d1ea06ffc7)
,
![{\textstyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bc926f90178739fccd01a96c6fa778ab3535d6)
, theorema praecedens ita exhibetur:
![{\displaystyle 2a\equiv {\frac {r}{q}}{\pmod {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6591c87345e112aa3e594d75b7656f5890fb65)
Quum quilibet factorum ipsius
![{\textstyle q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a32efd4de244331181b8fa97862015251f65da5f)
complementum suum ad
![{\textstyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b)
habeat in
![{\textstyle t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bc926f90178739fccd01a96c6fa778ab3535d6)
, erit
![{\textstyle q\equiv t{\pmod {p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/990770475c616f1172f6d3532bf9fcdd92143a79)
, quoties multitudo factorum par est, i.e. quoties
![{\textstyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b)
est formae
![{\textstyle 8n+1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef68fea94a1ff77877d042ef3ec14efdfbfd03a3)
, contra
![{\textstyle q\equiv -t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f58a18776bcc3f0d1a153e89de0319d3f658f13)
, quoties multitudo factorum impar est, sive
![{\textstyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad87bd7009e2a5c52bd0fb5a9bda9d8c1c23a79b)
formae
![{\textstyle 8n+5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55e0bda3351a362b5b64658249d9feca7b9244c7)
. Perinde in casu priori erit
![{\textstyle r\equiv s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c3f075839ca54e994540276be10b234ce5262c9)
, in posteriori
![{\textstyle r\equiv -s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2657acd217365e932edc14e4a53ddf186b4ca1dc)
. In utroque casu erit
![{\textstyle qr\equiv st}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c482c1c95e95304d85b0c4d3c4519f13aa7b3a1)
, et quum constet, haberi
![{\textstyle qrst\equiv -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/508e61946d3ea5d1b66ecf7b63f63c5923aa8c0b)
, erit
![{\textstyle qqrr\equiv -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88f6f8806267402fc38121d1666f6e747ffe3230)
,