Whether a prime number $ p $ can be written in the form $ 3A + 2B $, where $ A,B \in \mathbb{N} $.

Question

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.

Answer 1

This is an application of the Frobenius Coin Problem, whose solution is given by the following theorem.

Theorem Suppose that $ m,n \in \mathbb{N} $ satisfy $ \gcd(m,n) = 1 $. Then $ mn - m - n $ is the largest integer that cannot be written as $ ma + nb $, where $ a,b \in \mathbb{N}_{0} $.

Observe that $ 3A + 2B = 3(A - 1) + 2(B - 1) + 5 $. As $ \gcd(3,2) = 1 $, the theorem yields the following statements:

• Any integer $ > 3 \cdot 2 - 3 - 2 = 1 $ can be written as $ 3a + 2b $, where $ a,b \in \mathbb{N}_{0} $.

• Any integer $ > 1 $ can thus be written as $ 3(A - 1) + 2(B - 1) $, where $ A,B \in \mathbb{N} $.

• Therefore, any integer $ > 1 + 5 = 6 $ can be written as $ 3A + 2B $, where $ A,B \in \mathbb{N} $.

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 $’.