136 On a new Theorem concerning Prime Numbers. 
which satisfies the congruence pr! =1(modr). Since we know 
1 i! pie : 
maa ae aa wine eee — — . ] 
that aa Danger a pra +9 = 0 (to mod. yw) in place 
of the cycle 1, 2, 3,...7, we may obviously substitute the re- 
duced cycle 
r—l r7—o 
“9? ae “9? @oe 
r—3 r—l 
—1,0,1,..— “== 
2 2 
oodies 
Thus, ea. gr., , when pw is of the form 6n+ 1, 
aia 1 1 1 
Fay 8 + a go ee 
and when wp is of the form 6n—1, 
—1 1 1 1 
tee ee) Maen ae Ee 
When g is 2, the theorem which replaces the preceding is as 
.—1,to mod. p. 
Q-1_] 
follows: » when p is of the form 4n +1, 
: po 
1 1 1 1 a 1 
See eg eb fad ae ee ee 
foal pk p—d poo 5 
i} 1 1 
and when p is of the form 4n—1, 
a0), dalton oan Be aT ac 
— por lo pp? pd ph pS 
1 | sa il 
+ eae + re, + &c., to mod. p. 
When q is not a prime, a similar theorem may be obtained by 
the very same method, but its expression will be less simple. 
The above theorems would, I think, be very noticeable were it 
only for the circumstance of their involving (as a condition) the 
primeness as well of the base as of the augmented index of the 
familiar Fermatian expression g#~1,—a condition which here 
makes its appearance (as I believe) for the first time in the 
theory of numbers. 
