Find the smallest prime number such that....

First we show 41 can't be expressed in this form.

Case 1:

$2^a=41+3^b$ . Taking mod 8,we get $2^a=2,4$ mod 8.That implies a=1 or 2.Easy to see that's impossible.

Case 2:

$3^b=41+2^a$ .Again,taking mod 8,we see that,b has to be even.Otherwise,a has to be 1.Since,43 is not a power of 3,that's impossible.

Now,taking mod 3,we see a has to be even.letting $b=2b_1$ and $a=2a_1$ we see, $41=(3^{b_1}-2^{a_1})(3^{b_1}+2^{a_1})$ SInce,41 is a prime easy to see that this can't be true. Now other primes less than 41 (other than 2 and 3) can easily be found to be expressed in this form using trial and error.