Let $n=am+1$ where $a $ and $m>1$ are positive integers and let $p$ be the least prime divisor of $m$. Prove that if $a<p$ and $ m \ | \ \phi(n)$ then $n$ is prime.

This question is a generalisation of the question at Let $n=apq+1$. Prove that if $pq \ | \ \phi(n)$ then $n$ is prime.. Here the special case when $m$ is a product of two distinct odd primes has been proven. The case when $m$ is a prime power has also been proven here https://arxiv.org/abs/2005.02327.

How do we prove that the proposition holds for an arbitrary positive integer integer $m>1 $? ( I have not found any counter - examples).

Note that if $n=am+1$ is prime, we have $\phi(n)= n-1=am$. We see that $m \ | \ \phi(n) $. Its the converse of this statement that we want to prove i.e. If $m \ | \ \phi(n) $ then $n$ is prime.

If this conjecture is true, then we have the following theorem which is a generalisation ( an extension) of Lucas's converse of Fermat's little theorem.

**$\textbf {Theorem} \ \ 1.$**$ \ \ \ $ Let $n=am+1$, where $a$ and $m>1$ are positive integers and let $p$ be the least prime divisor of $m$ with $a<p$. If for each prime $q_i$ dividing $m$, there exists an integer $b_i$ such that ${b_i}^{n-1}\equiv 1\ (\mathrm{mod}\ n)$ and ${b_i}^{(n-1)/q_i} \not \equiv 1(\mathrm{mod}\ n)$ then $n$ is prime.

Proof. $ \ \ \ $ We begin by noting that ${\mathrm{ord}}_nb_i\ |\ n-1$. Let $m={q_1}^{a_1}{q_2}^{a_2}\dots {q_k}^{a_k}$ be the prime power factorization of $m$. The combination of ${\mathrm{ord}}_nb_i\ |\ n-1$ and ${\mathrm{ord}}_nb_i\ \nmid (n-1)/q_i$ implies ${q_i}^{a_i}\ |\ {\mathrm{ord}}_nb_i$. $ \ \ $${\mathrm{ord}}_nb_i\ |\ \phi (n)$ therefore for each $i$, ${q_i}^{a_i}\ |\ \phi (n)$ hence $m\ |\ \phi (n)$. Assuming the above conjecture is true, we conclude that $n$ is prime.

Taking $a=1$, $m=n-1$ and $p=2$, we obtain Lucas's converse of Fermat's little theorem. Theorem 1 is thus a generalisation (an extension) of Lucas's converse of Fermat's little theorem.

On recommendation by the users, this question has been asked on the MathOverflow site, https://mathoverflow.net/questions/373497/prove-that-there-are-no-composite-integers-n-am1-such-that-m-phin