All numbers not divisible by 3 can be represented as 3n±1, where n is an integer, simply because, of the two integers between each multiple of 3, one follows the multiple of 3 and the other precedes the next multiple.
If we add 2 we get either 3n+1+2=3n+3 (divisible by 3); or 3n-1+2=3n+1.
If 3n+1 is a prime number, then adding 2 to it will create a number divisible by 3.
If 3n-1 is a prime number, then adding 4 to it will create a number divisible by 3.
Therefore, with the exception of 3, 5, 7, you cannot have 3 successive odd numbers all of which are prime. And, of course, apart from 2, all even numbers are composite.