Example 4 Write down all three-term arithmetic sequences of prime numbers with a common difference of 8.
(2009, Tsinghua University Independent Admission Examination)
Let these three numbers be $a$, $a+8$, $a+16$.
Since the remainders when $a$, $a+8$, $a+16$ are divided by 3 are all different, one of these three numbers must be 3. And since $a+8$, $a+16$ are prime numbers greater than 3, it follows that $a=3$.
Therefore, the sequence of numbers is $3,11,19$.