3^m-2^m=65

To solve the equation \( 3^m - 2^m = 65 \), we can approach it by testing some values of \( m \).

Let's try a few integer values:

1. For \( m = 1 \): \[ 3^1 - 2^1 = 3 - 2 = 1 \quad (\text{not } 65) \]

2. For \( m = 2 \): \[ 3^2 - 2^2 = 9 - 4 = 5 \quad (\text{not } 65) \]

3. For \( m = 3 \): \[ 3^3 - 2^3 = 27 - 8 = 19 \quad (\text{not } 65) \]

4. For \( m = 4 \): \[ 3^4 - 2^4 = 81 - 16 = 65 \quad (\text{this works!}) \]

So, \( m = 4 \) is a solution.

To verify, substitute \( m = 4 \) back into the original equation: \[ 3^4 - 2^4 = 81 - 16 = 65. \]

Thus, the solution is: \[ \boxed{4}. \]