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

prime-numbers conjectures

Solution 1:

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.

Related

binomial congruence $\sum_{i=1}^{\frac{p-1}{2}}\binom{2i}{i}\equiv 0~or (-2)\pmod p$
Easiest way to show $x^2 - 229y^2 = 12$ has no solutions in integers
$\frac{1}{e}=$"Probability that every chocolate goes into a wrong spot".
Are column vectors considered the "default" orientation in linear algebra?
Using Stokes' theorem to define the exterior derivative operator
Zolotarev's Lemma and Quadratic Reciprocity
Venn diagram with rectangles for $n > 3$
How to prove $\lim_{x\rightarrow0^+}\sum_{k=1}^{+\infty}\frac{\sin{(x\sqrt{k})}}{k}=\pi$?
The $2$-category of monoids
Catalan numbers and triangulations
Proof of an inequality using analytic geometry
Ordering finite groups by sum of order of elements