Numeri primi, quorum residuum vel non-residuum est —1, facile dignos 
cuntur adiumento theorematis sequentis, quod etiam per se ipsum satis memo 
rabile est. 
Theorema. Productum e duobus factoribus 
W' = 1 —j— r 1 —|— -r 4 —etc. —|— (” 1 ) 2 
W — \ r-\-P etc. -J- ri w— T 
est = n, si n est impar ; vel — 0, si n est impariter par ; vel = 2 n, si n est 
pariter par. 
Demonstr. Quum manifesto fiat 
W=r-\-r*-\-r 9 -f- etc. -|-ri 
r 4 —j— r 9 —1— etc. -j-r^+i) 2 
r 9 4~ etc. etc. 
productum IdK' ita quoque exhiberi poterit 
1 —J— ^ —J— etc. 
_|_ r —i _(_ /* _j_ r 9 4- 4 6 —|— etc. —j— r nw ) 
-f r“ 4 (r 4 + r 9 ri- r 16 -f- r 25 + etc. + r {n+xY ) 
4_ r — 9 4° _j_ 4 6 4~ r 25 -f- r 36 4~ etc. 4~ ) 
etc, 
4_ fn-\y 4_ 4_ r (m+!) 2 4_ r («+ 2 ) 2 4_ etc _|_ r (2w- 2 ) 2 
quod aggregatum verticaliter summatum producit 
-f r(i-hrr + r 4 4- r ,! 4- etc. 4-r 2w-2 ) 
4-y 4 ( 14- r 4 -f- r 8 4- r 12 4~ etc. 4~ r in ~‘' A 
4-r 9 (i4-r 6 4-4 2 4- r 18 4~ etc. 4~r 6n 
4- etc. 
4-(i 4-r 2w ~ 2 4- P n 4 4-r Gn ~ 6 4- etc. -4 ) 
lam si n impar est, singulae partes huius aggregati, praeter primam n, erunt 
= 0; secunda enim manilesto fit \_ rr , tertia \zLf* etc. Quoties vero n 
par est, excipere insuper oportebit partem 
quae fit = 
= n-\-nr* m 
WW'= 2 n 
WW = 0. 
lam p 
vel = —1, 
priori esse d 
concludimus 
mae 4pt —f-1 
Deniq 
sponte seqm 
8 {i, 4~ 3 ; atq