Originally Posted by

**Bashyboy** Hello,

The exact question is, Prove or disprove that there are three consecutive odd positive integers that are primes, that is, odd primes of the form $\displaystyle p$,$\displaystyle p+2$, and $\displaystyle p+4$.

At first glance, I did not realize that this statement was an existentially quantified one, that one example would prove this true--the example being 3, 5, and 7. So, I did a full-fledged proof and am wondering if I did it correctly, notwithstanding its necessity.

Let $\displaystyle p$ be an odd integer that is prime, then $\displaystyle p = 2k + 1$, where $\displaystyle k \in \mathcall{Z}$.

p=

$\displaystyle p+2=(2k + 1)+2 \rightarrow p+2= 2k + 3$ this is prime, because the only factors are $\displaystyle 2k+3$ and 1.