I would like to know whether or not a prime number $ p $ can be written in the form $$ p = 3A + 2B, $$ where $ A $ and $ B $ are positive integers.
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This is an application of the Frobenius Coin Problem, whose solution is given by the following theorem.
Observe that $ 3A + 2B = 3(A - 1) + 2(B - 1) + 5 $. As $ \gcd(3,2) = 1 $, the theorem yields the following statements:
Notice that the numbers $ 1 $, $ 2 $, $ 3 $, $ 4 $ and $ 6 $ cannot be put in the required form, whereas $ 5 = 3 \cdot 1 + 2 \cdot 1 $ can. Therefore, the answer to your problem is ‘all prime numbers $ \geq 5 $’.