<s>[INST] 3, 5, 7 are the only three consecutive odd natural numbers that are prime. Can someone please help me proof this theorem. So far I assumed towards contradiction that there exist three consecutive odd natural numbers that are primes P, P+2, and P+4. 
Theorem: 3, 5, 7 are the only three consecutive odd natural numbers that are prime.
 [/INST] Let's get three numbers: $n, n+2, n+4$.
We have 2 cases:


*

*$n\not\equiv0\mod 3$: 


Thus $n=3k+1$ or $n=3k+2$, so $n+2=3(k+1)$ or $n+4 = 3(k+2)$


*$n=3k$


In both cases at least one of our numbers is divisible by $3$. 
The only prime that is divisible by $3$ is $3$.
The only sequention of prime numbers $n, n+2, n+4$ that contains $3$ is $3,5,7$.
So the only sequention of prime numbers $n, n+2, n+4$ is $3,5,7$.
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