Example. Find all arithmetic sequences of three prime numbers with a common difference of $\varepsilon$.

Find all arithmetic sequences of three prime numbers with a common difference of $\varepsilon$.
Solution: Clearly, these three prime numbers must be odd. If the first term is $2n+1$, then the three terms of this sequence are $2n+1, 2n+9, 2n+17$. Consider $n$ modulo 3:

If $n \equiv 0(\bmod 3)$, i.e., $n=3k$, then $2n+9=6k+9$, which is not a prime number;

If $n \equiv 1(\bmod 3)$, i.e., $n=3k+1$, then $2n+1=6k+3$, which is not a prime number when $k \neq 0$;

If $n \equiv 2(\bmod 3)$, i.e., $n=3k+2$, then $2n+17=6k+21$ is not a prime number.

Therefore, only when $k=0$, i.e., $n=1$, $2n+1, 2n+9, 2n+17$ can be prime numbers. At this time, the three prime numbers are 3, 11, 19.
Thus, the only solution to this problem is: $3, 11, 19$.