TMUA 2023 Paper 2 Question 4

There are two consecutive odd integers that are prime (for example: $17$, $19$).

Any three consecutive odd integers can be written in the form $n-2$, $n$, $n+2$ for some $n$.

If $n$ is one more than a multiple of $3$, then $n+2$ is a multiple of $3$.

If $n$ is two more than a multiple of $3$, then $n-2$ is a multiple of $3$.

The only other possibility is that $n$ is a multiple of $3$.

In each case, one of the integers is a multiple of $3$, so not prime.

Therefore the largest number of consecutive odd integers that are all prime is two.