There is a famous unsolved problem called Lehmer's Totient Problem which states that,
$\varphi(n)\mid n-1 \implies n$ is a prime.
Where $\varphi(n)$ is Euler's Totient Function.
I was wondering about two conjectures of my own which I found unable to prove or disprove myself. The conjectures are,
Conjecture 1
If $n$ is an odd integer then $\varphi(n)+1\mid n \implies n$ is a prime.
Conjecture 2
For odd $n$, $\varphi(n)+1\mid n\iff \varphi(n)\mid n-1$
Note that both the conjectures are independent of one another and the proof of any one willn't shed light on the proof of disproof the other unless a proof or disproof of Lehmer's original conjecture is done.
So, can anyone help me in proving anyone of the conjectures?
Note:-
I don't know whether my conjectures are already well-known conjectures or not. If that's the case then any reference to the authentic source(s) will be enough.
Added:-
The conjectures are now edited taking in view the suggestions of fretty and A. Nicolas.
The conjecture should be modified, since for example if $n=2p$, where $p$ is an odd prime, then $\varphi(n)+1$ divides $n$.