GULF 2017 Problem 1

1- Find a pair $(m,n)$ of positive integers such that $K = |2^m-3^n|$ in all of this cases : $a) K=5$ $b) K=11$ $c) K=19$ 2-Is there a pair $(m,n)$ of positive integers such that : $$|2^m-3^n| = 2017$$ 3-Every prime number less than $41$ can be represented in the form $|2^m-3^n|$ by taking an Appropriate pair $(m,n)$ of positive integers. Prove that the number $41$ cannot be represented in the form $|2^m-3^n|$ where $m$ and $n$ are positive integers

4-Note that $2^5+3^2=41$ . The number $53$ is the least prime number that cannot be represented as a sum or an difference of a power of $2$ and a power of $3$ . Prove that the number $53$ cannot be represented in any of the forms $2^m-3^n$ , $3^n-2^m$ , $2^m-3^n$ where $m$ and $n$ are positive integers