786 
expression for the polynomial 4 (et), we need only consider the case 
that m has no square divisors, and a further limitation is still possible. 
In fact, when m having a single factor 2 is equal to 22 we have 
and thus the construction of the primitive divisor in general is made 
to depend on the case that m is a product of unequal odd prime 
numbers. 
V. Let p be a prime number not dividing the integer n and 
m==pn, then 
F,(«) — 1 is divisible by mrt — 1, 
when ” is not a factor of pl. 
On the contrary. when p—1 =Án 
vw-1—] 
I " (x) 
F(a) — Lis divisible by 
4 
and 
Fale) — p is divisible by F,,(0). 
VI. Let p be a prime number not dividing the integer 1 and 
i= pn, then 
F(a) — a?) is divisible by a+! — 1, 
when ” is not a factor of p + 1. 
On the contrary, when p + 1 = kn 
Pek 3 apti—l1 
F(x) — ar“ is divisible by Ee 
| Y Fle) 
and 
Ent) + pare” is divisible by Mr). 
VII. The sum of the roots «, of the primitive divisor /,,(#) is 
equal to u (mm). 
VIII. Denoting by D the greatest common measure of the integers 
k and m and supposing m to have no square divisors the sum of 
the At powers of the roots w, is equal to 
ge (mm) a(D) ¢ (D). 
From the known values of the sums ak, == 41, 253-25 
ed 
coefficients A, of the polynomial 
Fn (ee) =A, + A,e + Aa? + ....4+ Auel! 
might be calculated, but we may proceed ina slightly different way. 
Supposing m to be a product of unequal odd prime numbers the 
integers r less than m and prime to m may be arranged into two 
