Prime numbers of the form : $2^{n+a}+2^{n} \pm 1$ , where $0 \leq a < n$ and $n \equiv 0 \pmod 6$

Question

Is it true that :

For any positive integer $n$ such that $n \equiv 0 \pmod 6$ there is at least one prime number of the form:

$p=2^{n+a}+2^{n} + 1$ , or , $p=2^{n+a}+2^{n} - 1$

with following properties : $0 \leq a < n$ , and $a\in \mathbf{Z^{*}} ; n\in \mathbf{Z^{+}} $

I have checked statement for each $n$ up to $n=1002$ and I haven't found any counterexample.

Any idea how to prove or disprove statement above without using a computer?

Answer 1

The conjecture appears to be false for $n=2184$.