<s>[INST] Consecutive Prime Gap Sum (Amateur) List of the first fifty prime gaps:
1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4.
My conjecture is that the sum of consecutive prime gaps is always prime whenever a prime gap of 2 is added.
$$ 1 + 2 = 3 $$
$$ 1 + 2 + 2 = 5 $$
$$ 1 + 2 + 2 + 4 + 2 = 11 $$
$$ 1 + 2 + 2 + 4 + 2 + 4 + 2 = 17 $$
$$ 1 + 2 + 2 + 4 + 2 + 4 + 2 + 4 + 6 + 2 = 29 $$
I don't know if this is meaningful or how to go about testing it completely (I've tested it up to 461) so I'll just leave this here and see what comes of it.
 [/INST] Let $p_n$ denote the $n^{\text{th}}$ prime. Then $p_{n+1}-p_n$ is the $n^{\text{th}}$ prime gap. The sum of the first $k$ prime gaps is $$\sum_{n=1}^k(p_{n+1}-p_n) = p_{k+1} - p_1 = p_{k+1} - 2.$$ Now, if the $k^{\text{th}}$ prime gap is $2$, that is $p_{k+1} - p_k = 2$, then $p_{k+1} - 2 = (p_k + 2) - 2 = p_k$ which is prime as you've noticed.
A pair of consecutive primes $p_n$, $p_{n+1}$ which differ by two (i.e. the $n^{\text{th}}$ prime gap is two) are said to be twin primes. It is still unknown as to whether there are infinitely map twin primes.
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