Question

Prove or Disprove that there are three consecutive odd positive integers that are primes, that is odd primes of the form $p$, $p+2$and$p+4$ .

Short answer

3, 5 and 7are primes for which the given statement is satisfied

Step-by-step solution

Step 1

An integer$p$larger than$x$is called prime if the only positive factors of$p$are 1 and$p$.

An integer that is not prime is called composite.

Step 2

To prove: There exist 3 consecutive and positive integers that are primes.

PROOF

3,5 and 7 are all primes, while they are also consecutive odd positive integers and thus 3, 5 and 7 are primes that satisfy the given statement.

Note: 3, 5 and 7 are the only consecutive odd positive integers that are primes, because every other sequence of 3 consecutive odd positive integers will have one integer that is divisible by 3.