Smallest number $n$ such that $1$ $\pmod n$ is not representable by sum of powers of $2$ and $3$

Question

Which is the smallest integer $n$ not divisible by $2$ or $3$ such that there does NOT exist two integers $x$ and $y$ such that $2^x+3^y$ $=$ $1$ $\pmod n$? If it is not $n = 683$ (the multiplicative orders for $2$ and $3$ $\pmod {683}$ are namely $22$ and $31$ so there will be few residue combinations to check.) Then is it true that there will always be solutions $x$ and $y$ to the following equation for any $n$ such that $\gcd(n, 6)$ $=$ $1$? If proving is too hard please find a counterexample $n$ relatively prime to $6$ such that there doesn't exists solutions to $2^x+3^y$ $=$ $1$ $\pmod n$. Thanks for help and suggestions on this question.

Answer 1

A brute force check shows that for $n=85$ there is no solution to $2^x+3^y \equiv 1 \pmod {85}$. Other values are $91, 341,425, 455, 511, 595, 637, 671, 935, 949$ They do not appear to be rare.