Alternative Form of Goldbach's Conjecture

The Goldbach Conjecture states that every number from 4 onwards can be written as a sum of primes in at least one way. It is equivalent to the statement that every number can be written as a sum of three primes.
For, suppose the Goldbach conjecture holds. Then for
$n \ge 4, \; 2n-2 \ge 6$
and so can be written as a sum of two odd primes and we can write
$2n-2=p_1+p_2$
. Then
$2n=p_1+p_1+2, \; 2n+1=p_1+p_2+3$
, showing that
$2n, \; 2n+1$
can be written as a sum of three primes and all integers from 8 onwards can be written in this form. We can also write
6=2+2+2, 7=2+2+3
Conversely suppose that every integer greater than or equal to 6 can be written as the sum of three primes. Consider the integer
$2n, n \ge 3$
. We know that
$2n+2=p_1+p_2+p_3$
.
$p_1, \; p_2, \; p_3$
cannot always all be odd, or we could not add them to obtain an even number. Hence one of them must be 2 - suppose
$p_1=2$
then
$2n=p_2+p_3$
. If one of
$p_2, \; p_3$
is 2 then so is the other since
$p_1+p_2$
$n \gt 3$
$2n$