Inspired by a conjecture of user Ante , I arrived at the following strengthened conjecture :
Let $\ m\ $ be a positive integer with $\ m\neq 3\mod 6\ $. Then, there exists a positive integer $\ n\ $ with $\ \gcd(m,n)=1\ $ and $\ \varphi(n)=\varphi(n+m)\ $
$\varphi(n)$ is the totient function.
For numbers of the form $\ 6k+3\ $ , I did not always find a solution, for example I found none for $\ m=9\ $.
The conjecture is true at least upto $\ m=\ 30\ 000\ $
Can we proof or disproof it ?