# Goldbach Partition - Why Co-Primality?

Any even number $2n$ can be written as the sum of two primes, $p_{a}$ and $p_{b}$. For $n \geq 2$ this is the Goldbach Conjecture.

$$p_{a} + p_{b} = 2n$$

Why are $p_a$ and $2n$ co-prime? That is, $p_a$ is not a factor of $2n$?

Because $p_a$ and $p_b$ are coprime to each other, they are also both coprime to their sum, $p_a{+}p_b$.
From the contradiction side: clearly if say $p_a$ divides $p_a{+}p_b$, it must also divide $p_b$.
• I'm assuming here that $p_a$ and $p_b$ are distinct, which the Goldbach Conjecture doesn't actually require, but is an interpretation that fits with your question. In general, the second line leads to the conclusion that if $p_a$ divides $p_a+p_b$, we must have $p_a=p_b$ (since they are prime). – Joffan Nov 25 '17 at 1:13
Actually, we can have $p_a = p_b$ not coprime to $2n$. For instance $3 + 3=6$ gives the Goldbach decomposition of $6$.
But if $p_a$ and $p_b$ are distinct they must be coprime, and hence both are coprime to their sum.