# Conjecture: Every prime number is the difference between a prime number and a power of $2$

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.

• Please try to reach more streamlined titles in your future questions, taking as example the modified version of the present title. – Did Jul 23 '16 at 11:20
• @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. – Lehs Jul 23 '16 at 11:24
• 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? – Did Jul 23 '16 at 11:26
• @Did. Now everything is clear. Thanks for the tip! I didn't notice that also you edited the title. – Lehs Jul 23 '16 at 11:40

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.