There are infinitely many odd numbers not expressible as the sum of a prime number and a power of $2$ Prove that there are infinitely many odd integers that are not expressible as the sum of a prime number and a power of two.
This is a difficult problem. Please give me some hints and some examples of this.

Below is the key passage from the paper Gerry linked to.
If $k$ is a nonnegative integer, then at least one of the following congruences holds:
$$
\begin{aligned} 
2^k&\equiv1\pmod3,\\ 
2^k&\equiv1\pmod7,\\ 
2^k&\equiv2\pmod5,\\ 
2^k&\equiv8\pmod{17},\\ 
2^k&\equiv2^7\pmod{13},\\
2^k&\equiv2^{23}\pmod{241}.
\end{aligned}
$$
[JL:
This is easy to see by checking, case-by-case, that irrespective of the residue class of $k$ modulo $24$ at least one of the above congruences holds. Observe that $2$ is of order $12$ modulo $13$ and of order $24$ modulo $241$. The authors of that paper did it by first observing that the relevant residue classes modulo $2,3,4,8,12$ and $24$ cover all the integers.]
Now 
consider 
the 
congruences 
$1\pmod3$, 
$1 
\pmod 
7$, 
$2 
\pmod 
5$, 
$8 
\pmod 
{17}$, 
$2^7 
\pmod 
{13}$, 
and 
$2^{23} 
\pmod 
{241}$. 
Since 
the 
moduli 
are 
pairwise 
relatively 
prime, 
there 
are 
infinitely 
many 
integers 
that 
satisfy 
all 
the 
congruences, 
by 
virtue 
of 
the 
Chinese 
Remainder 
Theorem. 
Now, 
if 
an 
odd 
integer 
$a$ 
satisfies 
all 
the 
congruences, 
then 
all 
the 
integers 
of 
the 
form 
$a-2^k$
are 
divisible 
by 
one 
of 
the 
moduli 
$3, 
7, 
5, 
17, 
13 $
or 
$241.$ 
It 
follows 
that 
$a-2^k$
is 
not 
prime 
and 
therefore 
$a$ 
does 
not 
have 
the 
form 
$a=2^k+p$.
