A question about prime-gaps whose sum is a power of $2$.

Question

I found this prime: $127163$ such that $127163+26$ is the next prime and $127163-6$ is the previous prime, I mean $127163-6$, $127163$ and $127163+26$ are three consecutive primes. As you can see, the sum of the gaps (the gap between the original prime and the previous prime plus the gap between the original prime and the next prime) is $32=26+6$, a power of $2$. Are there other examples of such primes for which the sum of the gaps is a power of $2$?

Answer 1

$31-2, 31, 31+6=29,31,37$ and $2+6=8=2^3$.

$53,59,61$,

$59,61,67$.

Answer 2

Yes: $3,5,7$. Consecutive primes with sum of gaps $2+2 = 4 = 2^2$. Here is the sequence of prime gaps. I think you can find many more examples with it.

Answer 3

There are a lot. Here are some early examples for each power of $2$ showing the three consecutive primes:

• $2^2: \qquad 3,5,7$ (only example this size)
• $2^3: \qquad 23,29,31$ (large number of examples this size)
• $2^4: \qquad 211,223,227$
• $2^5: \qquad 4751,4759,4783$
• $2^6: \qquad 33223, 33247, 33287$