[seqfan] A371164 and A106309

[israel at math.ubc.ca]

I see A371164 and A106309 have been merged.  A371164 was  	

Primes p such that the polynomial x^5 - x^4 - x^3 - x^2 - x - 1 is 
irreducible mod p.

while A106309 was originally

Primes that yield a simple orbit structure in 5-step recursions.

explained in the Comment: 

Consider the 5-step recursion x(k)=x(k-1)+x(k-2)+x(k-3)+x(k-4)+x(k-5) mod 
n. For any of the n^5 initial conditions x(1), x(2), x(3), x(4), and x(5) 
in Zn, the recursion has a finite period. When n is a prime in this 
sequence, all of the orbits, except the one containing (0,0,0,0,0), have 
the same length.

My problem is that I don't see how to prove that these are equivalent. 
Certainly if the polynomial is irreducible mod p, all the orbits will have 
the same length, namely the least positive integer k such that x^k-1 is 
divisible by p(x) = x^5 - x^4 - x^3 - x^2 - x - 1 over the integers mod p. 
But what if p is reducible, say p(x) = q(x) * r(x) mod p with q and r 
distinct, and the least positive integer k such that x^k - 1 is divisible 
by q(x) mod p is the same as the least k such that x^k - 1 is divisible by 
r(x) mod p. Then, unless I am mistaken, all orbits will have length k. Now 
I haven't found an example where this occurs, but I don't know of any 
reason why it shouldn't happen.

Can anyone enlighten me?

Cheers,
Robert