I noticed a simple pattern in prime numbers, specially twin primes.Using which I was able to prove that every twin prime greater than $(5,7)$ is of the form, $(a+30b,a+2+30b)$, where $a=11,17,29$ and $b$ is a nonnegative integer.
My proof is very elementary and this result is a bit obvious. But I couldn't find anything about this in my number theory textbooks or on the internet. Is this result already known ? If so, where can I read more about this? Can anyone tell if this is a result of a more general pattern?
I already know about Dirichlet's Theorem of Prime Numbers and how it implies that these series will produce infinitely many prime numbers. I apologize in advance if this isn't a proper question.
Edit: This is indeed part of a more general pattern. Thus this question is solved. Thanks to Noah Schweber and kingW3. For example, every cousin prime greater than $(3,7)$ is of the form $(a+30b,a+4+30b)$, $a=7,13,19$.
Such a general form can be shown for sexy primes and primes with higher gaps. Although using higher primorials would be more sensible for showing such general patterns for higher prime gaps.