Whether a Prime number greater than can be written as sum of a Prime number and $2^n$?
$P_2 = P_1 + 2^N$
Some Examples of this
No, this is not true, as you can discover by looking at the number 127. It cannot be expressed in the required form, and I believe there are more examples.
Edit: Another example is 331, since it is prime, yet all the numbers $331-2^n$ are composite for $n=1, 2, \dots 8$.
There is a paper available online by Zhi-Wei Sun in which he gives some background and further examples, and the amazing statement that the integer
$$M = 47867742232066880047611079$$
plus or minus a power of $2$ can never be a prime, although I am not sure if $M$ is itself a prime (although my computer thinks it is likely to be).