<s>[INST] Number of ordered Pairs satisfying $4^m-3^n=1$ Find the Number of ordered Pairs $(m,n)$ of positive integers satisfying $4^m-3^n=1$
Mt try:
Trivially $m=n=1$ satisfies
Let $m \gt 1$
$$4^m-3^n=(1+3)^m-3^n=1$$
$\implies$
$$3\binom{m}{1}+3^2\binom{m}{2}+3^3\binom{m}{3}+\cdots+3^m=3^n$$
Now since LHS is not a power of $3$ and RHS is, this is possible only when $m=1$
Hence the only ordered pair is $(1,1)$
is this the right way?
 [/INST] $4^{m} - 3^{n} = 1$ or $4^m-1=3^n$ 
$(2^m-1)(2^m+1)=3^n$ therefore $2^m-1$ and $2^m+1$ are perfect powers of 3
Let $2^m+1=3^x$ and $2^m-1=3^y$
$3^x-3^y=2 (x>y)$  therefore $3^y(3^{x-y} -1) = 3^0.2^1$
Hence $ y=0$ and $3^{x-y} -1 =2$ implying $x=1$
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