Just out of curiosity, is there any prime gap sequences that have been proven (or suspected) to never occur ? For example, I noticed that within the first million primes, there is no prime gap of $8$ followed by a prime gap of $14$, and neither $14$ followed by $8$.
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3You have to avoid hitting a complete set of residues modulo a prime $p$ (unless you include $p$ in the set). The only odd prime gaps are $p-2$ where $p$ is an odd prime. This is because if there is an odd gap, one of the numbers must be even, and therefore must be $2$. So there is no gap of size $7$. – Mark Bennet Oct 02 '19 at 03:09
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If two primes $> 3$ are separated by a gap of $8$, for example, they have the form $$6 k - 1, 6 (k + 1) + 1$$ for some integer $k$. The number $14$ larger than the latter is $$[6 (k + 1) + 1] + 14 = 3 (2k + 7),$$ which is composite.
Alternatively, for any integer $k$, the integers $k, k + 8, k + 22$ have different residues modulo $3$, so (exactly) one of them is divisible by $3$.
Travis Willse
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