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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.

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    $\begingroup$ Assuming I didn't make a mistake every twin prime pair except of the $(5,7)$ is of the form $(210b+a, 210b+a+2)$ for $a=11,17,29,41,59,71,101,107,137,149,179,191,197,209$. I've picked $210$ because it's a primorial. $\endgroup$ – kingW3 Apr 23 '17 at 15:00
  • $\begingroup$ Wow, that's actually really helpful, thanks. $\endgroup$ – Saad Hasan Apr 23 '17 at 15:56
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Yes, this is part of a more general pattern. Ultimately, it just boils down to the observation that for any $a$ not of that form, either $a$ or $a+2$ shares a factor with $30$. This can be generalized straightforwardly to show similar limitations on arbitrary finite difference patterns in the primes. I don't know a citation for this, but this sort of argument is quite standard in number theory.

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  • $\begingroup$ Thank you for your answer. But I don't get how one could show a similar result for arbitrarily finite difference patterns in the primes by just using this observation. By using the same method I could only prove up to a prime gap of 24. Could you please elaborate a bit? $\endgroup$ – Saad Hasan Apr 23 '17 at 15:08
  • $\begingroup$ No need. I think I've got it. This can be shown by using primorials. $\endgroup$ – Saad Hasan Apr 23 '17 at 16:08

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