Pagina:Werkecarlf02gausrich.djvu/107

Primo saltem aspectu nexum simplicem inter valores numerorum ${\textstyle a}$, ${\textstyle b}$, quibus idem complexus respondet, non animadvertimus. At si perpendimus, diiudicationem similem in theoria residuorum quadraticorum per regulam simpliciorem absolvi respectu numeri ${\textstyle -3}$, quam respectu numeri ${\textstyle +3}$, spes affulget successus aeque secundi in theoria residuorum biquadraticorum. Invenimus autem, ${\textstyle -3}$ pertinere ad complexum

$${\begin{array}{c||c||c||c}A{\text{ pro}}&B{\text{ pro}}&C{\text{ pro}}&D{\text{ pro}}\\{\begin{array}{r|r|r}p&a&b\\37&+1&-6\\61&+5&-6\\157&-11&-6\\193&-7&-12\\&&\\&&\\&&\\&&\\&&\\&&\\\end{array}}&{\begin{array}{r|r|r}p&a&b\\5&+1&+2\\17&+1&-4\\89&+5&-8\\113&-7&-8\\137&-11&-4\\149&-7&+10\\173&+13&+2\\233&+13&+8\\257&+1&-16\\281&+5&+16\\\end{array}}&{\begin{array}{r|r|r}p&a&b\\13&-3&+2\\73&-3&-8\\97&+9&+4\\109&-3&+10\\181&+9&+10\\229&-15&+2\\241&-15&-4\\277&+9&+14\\&&\\&&\\\end{array}}&{\begin{array}{r|r|r}p&a&b\\29&+5&+2\\41&+5&-4\\53&-7&+2\\101&+1&+10\\197&+1&-14\\269&+13&+10\\293&+17&+2\\&&\\&&\\&&\\\end{array}}\end{array}}$$

ubi lex inductionis sponte se offert. Scilicet pertinet ${\textstyle -3}$ ad complexum

${\textstyle A}$, quoties ${\textstyle b}$ per 3 divisibilis est, sive ${\textstyle b\equiv 0{\pmod {3}}}$
${\textstyle B}$, quoties ${\textstyle a+b}$ per 3 est divisibilis, sive ${\textstyle b\equiv 2a{\pmod {3}}}$
${\textstyle C}$, quoties ${\textstyle a}$ per 3 est divisibilis, sive ${\textstyle a\equiv 0{\pmod {3}}}$

${\textstyle D}$, quoties ${\textstyle a-b}$ per 3 divisibilis est, sive ${\textstyle b\equiv a{\pmod {3}}}$

26.

Numerum ${\textstyle +5}$ adscribendum invenimus complexui

${\textstyle A}$ pro ${\textstyle p=101}$, ${\textstyle 109}$, ${\textstyle 149}$, ${\textstyle 181}$, ${\textstyle 269}$
${\textstyle B}$ pro ${\textstyle p=13}$, ${\textstyle 17}$, ${\textstyle 73}$, ${\textstyle 97}$, ${\textstyle 157}$, ${\textstyle 193}$, ${\textstyle 197}$, ${\textstyle 233}$, ${\textstyle 277}$, ${\textstyle 293}$
${\textstyle C}$ pro ${\textstyle p=29}$, ${\textstyle 41}$, ${\textstyle 61}$, ${\textstyle 89}$, ${\textstyle 229}$, ${\textstyle 241}$, ${\textstyle 281}$

${\textstyle D}$ pro ${\textstyle p=37}$, ${\textstyle 53}$, ${\textstyle 113}$, ${\textstyle 137}$, ${\textstyle 173}$, ${\textstyle 257}$

In considerationem vocatis valoribus numerorum ${\textstyle a,b}$ singulis ${\textstyle p}$ respondentibus, lex hic aeque facile, ut pro classificatione numeri ${\textstyle -3}$, prehenditur. Scilicet incidimus in complexum