Goldbach Conjecture implies Goldbach's Marginal Conjecture

Theorem

Suppose the Goldbach Conjecture holds:

Every even integer greater than $2$ is the sum of two primes.

Then Goldbach's Marginal Conjecture follows:

Every integer greater than $5$ can be written as the sum of three primes.

Proof

Suppose the Goldbach Conjecture holds.

Let $n \in \Z$ such that $n > 5$.

Let $n$ be an odd integer.

Then $n - 3$ is an even integer greater than $2$.

By the Goldbach Conjecture:

$n - 3 = p_1 + p_2$

where $p_1$ and $p_2$ are both primes.

Then:

$n = p_1 + p_2 + 3$

As $3$ is prime, the result follows.

Let $n$ be an even integer.

Then $n - 2$ is an even integer greater than $3$ and so greater than $2$.

By the Goldbach Conjecture:

$n - 2 = p_1 + p_2$

where $p_1$ and $p_2$ are both primes.

Then:

$n = p_1 + p_2 + 2$

As $2$ is prime, the result follows.

$\blacksquare$