56 
THEOREMATIS FUNDAMENTALIS IN DOCTRINA DE RESIDUIS QUADRATICIS 
täte diminutae divisibiles erunt per I — ; quamobrem in hoc casu etiam 
fix n )—p per 1 — x p et proin etiam per ~~ divisibilis erit. 
3. 
Theorema. Statuendo 
x — x* -f- x rm — x* 3 -f- x a * — etc. —- x aP 2 = 4 
i . • ■ ■ . ■ . 
erit 44 + p divisibilis per accepto signo superiori, quoties p est formae 
4 k —|— 1, inferiori, quoties p) est formae Ak-f-3. 
Demonstr. Facile perspicietur, ex p — 1 functionibus hisce 
-\-xi—xx-\-x a+1 — xf m+l -f- etc. -\~x aP ~ +1 
— x a i — x %0 ' -f- —<r* 3+a —}— etc. —x* P 1+a 
-f x rm 4 — x^ aa +x a3+aa — x a * +aa + etc. + x aP+aa 
— x°4 — x' ,j3 -f- x a * +a¡> — x a * +rjS + etc. -f- x aP+l+aS 
etc. usque ad 
x 
' £ ^ 
C, — X 
+<3? a 
i +* P ~ 2 _ aJ * p + 
primam fieri =0, singulas reliquas autem per 1 — x v divisibiles. Quare per 
1 — x p etiam divisibilis erit omnium summa, quae colligitur 
= ii—(/M — i) + (/+ +1 ) 
= 44— f(xx) +/++ 1 ) —/(<37 
-1) - (/(* aa+1 ) -1)+C/K 8+1 ) -1) - 
+ № a ^ 2+1 )—i) 
aa+1 ) -f-/(a? a * +1 ) — etc. -J- f{x aP '+ 1 ) = 12 
etc. 
Erit itaque haecce expressio Q etiam divisibilis per —f?-. lam inter exponentes 
2, (X-j-1, aa-f-1, ct 3 —1 a p ' + 1 unicus tantum erit divisibilis per p, 
puta a~^ p 1 ^ —f-1, unde per art. praec. singulae partes expressionis 12 hae 
f{xx'), /+ +1 ), f{x m+l ), (/++ 1 ) etc. 
excepto solo termino f{x ai{P ~ >]+1 ), divisibiles erunt per \=-f. Istas itaque par 
tes delere licebit, ita ut per etiam divisibilis maneat functio