<s>[INST] Is $53$ expressible in this form? It seems as if prime numbers may always be expressed in the form $a\cdot 2^b+c \cdot 3^d$ for some nonnegative integers $b,d$ and $a,c\in \{-1,0,1\}$.
Examples:
$2=1\cdot 2^1+0\cdot 3^d$
$3=0\cdot 2^b+1\cdot 3^1$
$5=1\cdot 2^1+1\cdot 3^1$
$7=1\cdot 2^2+1\cdot 3^1$
$11=1\cdot 2^3+1\cdot 3^1$
$13=1\cdot 2^2+1\cdot 3^2$
$17=1\cdot 2^3+1\cdot 3^2$
$19=1\cdot 2^4+1\cdot 3^1$
$23=1\cdot 2^5+(-1)\cdot 3^2$
$29=1\cdot 2^5+(-1)\cdot 3^1$
$31=1\cdot 2^5+(-1)\cdot 3^0$
$37=1\cdot 2^6+(-1)\cdot 3^3$
$41=1\cdot 2^5+1\cdot 3^2$
$43=1\cdot 2^4+1\cdot 3^3$
$47=1\cdot 2^7+(-1)\cdot 3^4$
We hit a brick wall at $53$. Can anyone confirm if $53$ is/isn't expressible?
What about $n\in \mathbb{N}$ in general? Is it possible to always express $n$ in this form?
Thanks.
EDIT: I invented this question on my own, there are no sources.
Note: The reason I started with prime numbers is because I found it hard to find an expression for $6$, whereas prime numbers continued to be easy to find expressions for (easy until $53$, that is).
 [/INST] You can check that for any $k,l$ we have that
$$2^k \not\equiv 3^l+53 \pmod{117}$$ 
and that
$$2^k+53 \not\equiv 3^l\pmod{117}$$ 
Just enumerating the different powers. 
Edit: If you consider the sets $Q_t = \{2^{k_1}3^{k_2}\dots p_t^{k_t}, k_1,\dots,k_t \ge 0\}$ and ask for integers not in $Q_t$ or sum or difference of elements in $Q_t$ then I found using this method that 103 is the smallest using primes up to 3 and that 583 is the smallest using primes up to 5 and I conjecture that 3737, 16579 and 41969 are the least integers using primes up to 7, 11 and 13 resp. 
It seems probable  that there is always an integer that can't be written in this way for any set of primes but I'm not sure how to prove it. 
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