Example 5. Does there exist a prime number that remains prime when 16 and 20 are added to it? If so, can the number of such primes be determined?
Solution: Testing with prime numbers $2, 3, 5, 7, 11, 13 \cdots$, we find that 3 meets the requirement. Below, we use the elimination method to prove that apart from 3, no other prime number satisfies the requirement.

Divide the natural numbers into three categories: $3n, 3n+1, 3n+2$ $(n \in \mathbb{N})$,
$\because 2n+1+20 = 2(n+7)$ is a composite number,
$3r+2+16 = 3(n+6)$ is a composite number,
thus numbers of the form $3n+1$ and $3n+2$ do not become prime numbers when 16 and 20 are added.

Among the remaining numbers of the form $3n$, it is clear that except for 3, when $n>1$, they are all composite numbers.
Therefore, the only prime number that meets the condition is 3.