Smallest Prime Number not Difference between Power of 2 and Power of 3

Theorem
$41$ is the smallest prime number which is not the difference between a power of $2$ and a power of $3$.

Proof
First we have:

$41 = 2^n - 3^m$.

We have that $n > 3$.

Thus:
 * $2^n \equiv 0 \pmod 8$

and as:
 * $41 \equiv 1 \pmod 8$

we have:
 * $1 \equiv -3^m \pmod 8$

which is not possible:

For any integer $k$,

So for any integer $m$, $1 \not \equiv -3^m \pmod 8$.

Now suppose $41 = 3^m - 2^n$.

We have that $m > 1$ and $n > 2$.

Taking $\mod 3$ on both sides:

which shows that $n$ is even.

Taking $\mod 4$ on both sides:

which shows that $m$ is even as well.

Now we take $\mod 5$ on both sides:

But $\paren {-1}^{m / 2} - \paren {-1}^{n / 2}$ can only be $0$ or $\pm 2$, not $1$, which is a contradiction.

Therefore $41$ is not the difference between a power of $2$ and a power of $3$.