Conjecture:
For each $n\in\mathbb N$ there are primes $q<p$ with $p-q=2^n$.
Verified for $n\leq 26$:
n p q
0 3 2
1 5 3
2 7 3
3 11 3
4 19 3
5 37 5
6 67 3
7 131 3
8 263 7
9 523 11
10 1031 7
11 2053 5
12 4099 3
13 8209 17
14 16421 37
15 32771 3
16 65539 3
17 131101 29
18 262147 3
19 524341 53
20 1048583 7
21 2097169 17
22 4194371 67
23 8388619 11
24 16777259 43
25 33554473 41
26 67108961 97
Proofs or counterexamples may be far away, but is something known about this topic?