Are there finitely many primes $x$ such that for a fixed odd prime $p$, $n=x^{p-1}+x^{p-2}+\dotsb + x+1$ is composite and $x \mid \phi(n)$?

Question

Let \begin{equation} n =x^{p-1}+x^{p-2}+\dotsb + x+1 \end{equation} where $x$ and $p$ are odd primes.

If $p$ is set to $5$, it appears $x=5$ is the only prime $x$ such that $n$ is composite and $x \mid \phi(n) $ (verified up to $x \le 2\cdot 10^6$). Setting $p$ to $7$, we get only two values of $x$; $x=7$ and $x = 281$.

If $x$ is allowed to take composite values, it appears there are infinitely many $x$ such that $x \mid \phi(n)$. Therefore, the following Conjecture is reasonable.

Conjecture 1

Let \begin{equation} n =x^{p-1}+x^{p-2}+\dotsb + x+1 \end{equation} where $x$ and $p$ are odd primes. For a fixed odd prime $p$, there are finitely many primes $x$ such that $x \mid \phi(n) $ with $n$ composite.

If Conjecture 1 is true, then, for a fixed prime $p$, there exists an upper bound $x_\text{max}$ such that $x \nmid \phi(n) $ for all $x>x_\text{max} $ with $n$ composite.

If Conjecture 1 is true, Theorem 1 gives a fast primality test for integers $x^{p-1}+x^{p-2}+\dotsb + x+1$ with $x > x_\text{max}$.

Theorem 1

Assuming Conjecture 1. Let $n =x^{p-1}+x^{p-2}+\dotsb + x+1$ where $x$ and $p$ are odd primes with $x>x_\text{max}$. If there exists a positive integer $b$ such that $b^{n-1} \equiv 1 \pmod n$ and $b^{(n-1)/x} \not\equiv 1 \pmod n$ then $n$ is prime.

Proof. Assuming Conjecture 1, we have $ x \mid \phi(n) $ if and only if $n$ is prime. Assume $n$ is composite. Using the properties of order of an integer, one can deduce that $ord_nb \mid (n-1) /x$. It follows that if $n$ is composite then $b^{(n-1)/x} \equiv 1 \pmod n$, contradicting our hypothesis. Therefore $n$ must be prime.

Note: As $x$ is prime then $x \mid \phi(n) $ implies $n = (ux+1)(vx+1)$ for some non negative integers $u$, $v$ with $ux+1$ prime. It can also be shown that $n = (sp+1)(tp+1)$ for some non negative integers $s$, $t$ with $sp+1$ prime. And from this post Positive divisors of $P(x,n)=1+x+x^2+ \cdots + x^n$ that are congruent to $1$ modulo $x$, if $ux+1 \mid n$ then $ux+1 \mid \frac{ u^{p}+1} {u+1}$. Perhaps these observations might be useful in proving Conjecture 1 and establishing $x_\text{max}$.

Answer 1

Here's a proof of Conjecture 1 for the case $p=5$. The proof depends on the truth of the following overwhelmingly true unproven result :

Let \begin{equation} P(x) = x^4 +x^3 +x^2 +x+1. \end{equation} Then all positive integers $x$ such that $P(x) $ has a proper divisor congruent to 1 modulo $x$ are given as follows :

Let $r(m)=m^2 +m-1$ and $q(m) = (r(m) +2)^2 - 2$, where $m$ is a positive integer. For a particular positive integer $m$, define sequences $A_n, B_n$ as follows:

$A_1 = m^3+2m^2 +2m$ , $A_2 = q(m)A_1+r(m)$, $A_n = q(m)A_{n-1}-A_{n-2}+r(m)$

$B_1 = m^5+2m^4 +3m^3 + 3m^2 +m$ , $B_2 = q(m)B_1+r(m)-m$, $B_n = q(m)B_{n-1}-B_{n-2}+r(m)$

Then all positive integers $x$ such that $P(x) $ has a proper divisor congruent to 1 modulo $x$ are given by $x=A_n$ and $x=B_n$, $n\ge 1$, $m \ge 1$.

It can be shown by induction that $A_n$ and $B_n$ are always composite except when $m=1$ and $n=1$ in which case $x=A_1=5$ is prime.

If the unproven result here is true (very likely the case), then Conjecture 1 is settled for the case $p=5$. (Having no proper divisor congruent to 1 modulo $x$ for all primes $x>5 $ implies $x \nmid \phi(P(x)) $ for all composites $P(x) $, $x >5 $ prime)

There's no reason to believe that the case $p=5$ is special. Conjecture 1 is most likely true for all odd primes.

ADDED (Compositeness of $A_n$ and $B_n$)

For $n \equiv 0, 1 \ (\mathrm{mod} \ 3)$, it can be shown by induction that $m$ divides $A_n$ and $B_n$. And when $n \equiv 2 \ (\mathrm{mod \ 3})$, one can prove that $m+1$ divides $A_n$ and $B_n$. Therefore it's clear that $A_n$ and $B_n$ are composite if $m>1$.

The remaining case $m=1$ is interesting. When $m=1$, $A_n$ and $B_n$ are a product of two sequences i.e $A_n = f_n\cdot g_n$ where $f_1=1 , f_2 = 3, f_n = 3f_{n-1}-f_{n-2}$ and $g_1=5 , g_2 = 12, g_n = 3g_{n-1}-g_{n-2}$.

Similarly $B_n = f_n\cdot g_n$ but with initial values changed; $f_1 = 2, f_2= 5$ and $g_1 = 5, g_2 = 14$

So $x = A_1 = 5$ is the only prime $x$ such that $P(x) =x^4 +x^3 +x^2 +x+1 $ contains a proper divisor congruent to $1$ modulo $x$