Does $1+x+x^2 ... x^{p -1}$ being a prime number imply p is prime?

Question

Let $p$ and $x$ be two positive integers greater than 2. If it is given that the sum : $$1+x+x^2+x^3... x^{p -1}$$ is a prime, is it possible to prove or disprove that $p$ is prime? If so, what would this proof be?

Answer 1

Assuming that you are asking about the sum $$ N=1+x+x^2+\cdots+x^{p-1}=\sum_{k=0}^{p-1}x^k, $$ then it is easy to see that this is composite, if $x>2$ and $p$ is composite. This follows from the geometric sum formula (it also telescopes nicely) $$ N(x-1)=x^p-1. $$ If $p=ab$ with $a>1,b>1$, then $$ N(x-1)=(x^a-1)(x^{a(b-1)}+x^{a(b-2)}+\cdots+x^a+1). $$ Here that extraneous $x-1$ is a factor of $x^a-1$ such that $(x^a-1)/(x-1)>1$. Therefore $N$ is composite.