Is every integer odd $p$ not divisible by $3$ a sum of difference of powers of $2$ and $3$ or vice versa? Does every odd integer $p$ not divisible by $3$ have the form 
$2^a+3^b$, $2^a-3^b$, and or $3^b-2^a$ for integers $a$ and $b$?
Please show proofs if possible. Thanks for help.

It can be verified by a finite if tedious computation that none of the equations
\begin{align}
  2^a + 3^b \equiv 133 \pmod {683} \\
  2^a - 3^b \equiv 133 \pmod {683} \\
  3^b - 2^a \equiv 133 \pmod {683}
\end{align}
have a solution $(a,b)$, and therefore $133$ cannot be written as sum or difference of a power of $2$ and a power of $3$.
(I chose $683$ because both $2$ and $3$ have relatively small multiplicative orders modulo $683$: $2^{22} \equiv 3^{31} \equiv 1 \pmod{683}$. Then I tested the above congruences by brute force for $0 \le a < 22$ and $0 \le b < 31$. I tried this for several primes, but $683$ was the first that worked.)
