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

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

which is not possible:
 * $\left({-3}\right)^{2 k} \equiv 1 \pmod 8$
 * $\left({-3}\right)^{2 k + 1} \equiv -3 \pmod 8$

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

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

Reducing $\bmod 3$ and $\bmod 4$ shows $m$ and $n$ are even.

But reducing $\bmod 5$ gives:
 * $1 \equiv \left({-1}\right)^{m / 2} - \left({-1}\right)^{n / 2}$

which is a contradiction.