Prime number in a polynomial expression

Question

Will be glad for a little hint: let x and n be positive integer such that $1+x+x^2+\dots+x^{n-1}$ is a prime number then show that n is prime

Answer 1

Hint $\ $ The sequence $\rm\:f_n = (x^n-1)/(x-1)\:$ is a divisibility sequence, i.e. $\rm\:m\:|\:n\:$ $\Rightarrow$ $\rm\:f_m\:|\:f_n.\:$

In fact it is a strong divisibility sequence, i.e. $\rm\:(f_m,f_n) = f_{\:\!(m,n)},\:$ which implies an intimate relationship between divisibility properties of the $\rm\:f_n\:$ and integers $\rm\:n\:$ (so, in particular, relation between notions associated with divisibility, such as irreducible = prime).

Answer 2

Let $n = kl$. Consider $$ (1 + x + ... + x^{k-1})(1 + x^k + x^{2k} + ... + x^{(l-1)k}) = (1 + x + ... + x^{n-1}) $$

Answer 3

Assuming that you meant that $1+x+x^2+\ldots+x^{n-1}$ is prime, note that this sum is $\frac{x^n-1}{x-1}$. If $n=ab$, we have $$\frac{x^n-1}{x-1}=\frac{(x^a)^b-1}{x-1}=\frac{(x^a-1)(1+x^a+x^{2a}+\ldots+x^{(b-1)a})}{x-1}\;,$$ and $\dfrac{x^a-1}{x-1}=\;$?