2010/004) Let x and n be positive integers such that 1 + x + x^2 + x^3 + ... + x^n-1 is a prime number.

Let x and n be positive integers such that 1 + x + x^2 + x^3 + ... + x^n-1 is a prime number. Show that n?
is a prime.

Proof:
1 + x + x^2 + x^3 + ... + x^n-1 = (x^n-1)/(x-1)

now in case n is not a prime then say n= pq

now x^n-1 = (x^(pq)-1) = (y^q-1)/(x-1) where y = x^p
= (y-1)/(x-1)(y^(q-1) + y^(q-2) + ...1)
= (x^p-1)/(x-1) (y^(q-1) + y^(q-2) + ...1)
= (x^p-1 + +1)((y^(q-1) + y^(q-2) + ...1)
has got 2 factors and hence not a prime
so n must be prime