Showing the equivalence of two forms of the Goldbach Conjecture

(Goldbach $\implies$ Euler)

Suppose that $n\geq 6$ (i.e. $n>5$) is even. Express $n$ as a sum of three primes $$n=p_1+p_2+p_3.$$ What do you think one of the primes must be, if we're going to conclude that any even integer $\underline{\,\,\,\mathbf{\geq 4}\,\,\,\,\,}$ is a sum of two primes? Modular arithmetic will help you prove that this is one of the three primes.

(Euler $\implies$ Goldbach)

Suppose that $n\geq 6$. Consider $n-2$ or $n-3$ depending on whether $n$ is even or odd.

HINT:

Let us consider the first few numbers considered in both the weak and strong form. For n = 6, they both say that it can be hit. For n = 7, the weak form says it can be hit, and it's 3 more than 4. For 8, both. For 9, the weak says it can be hit, and it's 3 more than 6 (an even number).

That's funny. 3's a prime, isn't it?