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Conjecture:

$\forall p\in\mathbb P\exists q\in\mathbb P\exists n\in \mathbb N: q-p=2^n$

Verified for the 100 first primes.

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    $\begingroup$ Please try to reach more streamlined titles in your future questions, taking as example the modified version of the present title. $\endgroup$ – Did Jul 23 '16 at 11:20
  • $\begingroup$ @Did, do you mean that the present title is bad or good? Are you refering to other of my titles? It's difficult to understand what you're aiming at. $\endgroup$ – Lehs Jul 23 '16 at 11:24
  • $\begingroup$ Seriously? What I wrote explicitely is that the former title was bad, that I modified it, and that you could try to imitate the current title for your future questions on the site. Clearer now? $\endgroup$ – Did Jul 23 '16 at 11:26
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    $\begingroup$ @Did. Now everything is clear. Thanks for the tip! I didn't notice that also you edited the title. $\endgroup$ – Lehs Jul 23 '16 at 11:40
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This question discusses the existence/infinitude of primes $p$ that can be written in the form $$p = q \pm 2^n$$ where $q$ is a prime and $n \in \mathbb{Z}^+$.

In particular for $p = q - 2^n$, Gjergji Zaimi mentions in a comment to his asnwer that $$p = 47,867,742,232,066,880,047,611,079$$ is a counterexample.

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