[Math] Prove or disprove that there are $n$ consecutive odd positive integers that are prime

discrete mathematics

Question:

Prove or disprove that there are $n$ consecutive odd positive integers that are prime.

If my answer for the question above is correct, then a new question arises.

My Attempt:

Odd numbers consist of multiples of $5$. I think that address the question.

New Question:

Is there at most $3$ consecutive primes? If So how would someone tackle this?

Best Answer

E.g. $17,19,21,23$ are $4$ consecutive odd numbers, none of which are divisible by $5$. So the observation implies the maximum number of consecutive odd numbers that are primes is at most $4$ (and we should be careful: there is one prime that ends in $5$).

We know $3,5,7$ are three consecutive odd numbers that are primes. So we next have one of two tasks:

  • Find an example of four consecutive odd numbers that are primes; or
  • Prove that no such example exists.

It's natural to consider the numbers $a,a+2,a+4$ modulo $3$ (to see if we find a factor of $3$).

  • If $a \equiv 0 \pmod 3$, then ...?
  • If $a \equiv 1 \pmod 3$, then ...?
  • If $a \equiv 2 \pmod 3$, then ...?
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