examday

/TMUA

A student attempts to answer the following question.

What is the largest number of consecutive odd integers that are all prime?

The student's attempt is as follows:

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

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

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

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

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

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

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

Which of the following best describes this attempt?