TITLE: How to disprove that every odd number can be written in the form $2^n + p$ with $p$ prime?
QUESTION [8 upvotes]: How can I disprove that every odd number, $2k+1>1$ 
can be written in the form $2k+1 = 2^n + p$ with $p$ prime?
I know it's not true but I don't know how to explain that it is not true.

REPLY [11 votes]: It suffices to find a counterexample.
After some searching, we find A133122

Odd numbers which cannot be written as the sum of an odd prime and a power of two
$$1, 3, 127, 149, 251, 331, 337, 373, 509, 599, 701,\dots$$

Even allowing $n=0$ and the use of even primes to say $3=2^0+2$ and ignoring $1$, the smallest counterexample is apparently $127$.
To prove that $127$ is in fact a counterexample, note that $127 = 64+3^2\cdot 7 = 32+5\cdot 19 = 16 + 3\cdot 37 = 8+7\cdot 17=\dots$ and so no power of two is valid.
These numbers are named Obstinate Numbers.