Closely related to this :
Conjecture about the totient function
$\varphi(n)$ is again the totienr function.
If $\ m\ $ is a positive integer coprime to $\ 15\ $ , there exists a positive integer $\ p\ $ such that $\ \varphi(pm+1)=\varphi(pm+m+1)\ $ , hence we can choose the desired number $\ n\ $ even to be congruent to $\ 1\ $ mod $\ m\ $.
Since some solutions are very large, I consider to also check the cases that $\ m\ $ is not coprime to $\ 15\ $ , for which I sometimes did not find a solution, but maybe just because I gave up too early.
The conjecture is true upto at least $\ m=427\ $