20
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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?

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  • 3
    $\begingroup$ This follows from Shinzel's Hypothesis H. Indeed that hypothesis would show that every even number can be written as the difference of two primes. Of course, little is known by way of actual proofs of specific cases. $\endgroup$
    – lulu
    Jul 16, 2016 at 16:16
  • 2
    $\begingroup$ According to this Chen's work on the Goldbach conjecture also implies that every (sufficiently large?) even number is the difference between a prime and a $P_2$ (a product of two primes). $\endgroup$
    – lulu
    Jul 16, 2016 at 16:21
  • 2
    $\begingroup$ This is a special case of an open problem,that every even number can be written as a difference of two primes.I believe it is true,but it seems to be an extremely difficult problem. $\endgroup$ Jul 16, 2016 at 18:00
  • 3
    $\begingroup$ Some more terms at OEIS A056206. $\endgroup$ Jul 16, 2016 at 23:13
  • 2
    $\begingroup$ This is a great question. $\endgroup$ Jul 18, 2016 at 18:59

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