Based on experimental data while analyzing the related question posted here I observed the following which I am presenting as conjectures.
Conjecture 1: For every natural number $k$, the equation $\varphi(x) = \varphi(x+2k)$ has infinitely many solutions. Further, if $x = n$ is a solution then $\frac{n}{\varphi(n)}$ is bounded.
Conjecture 2: The only solutions of $\varphi(x) = \varphi(x+3)$ are $x = 3,5$.
$\varphi(n)$ denotes the totient function.
Any reference to these in literature.
Update: I have verified conjecture 2 upto $n = 2 \times 10^8$.
Best Answer
Partial answer :
Conjecture $2$ is false : One of the entries in the OEIS-sequence in the linked question besides of $1$ is congruent $1$ modulo $3$. With the help of this entry I constructed $$n=9134280520365$$ which satisfies $$\varphi(n)=\varphi(n+3)$$