Goldbach Conjecture equivalency

Hi,

Playing around with the Goldbach conjecture (that an even number is a
sum of two primes) I came to the interesting and I would assume more
general conjecture that provided no prime is 2, the sum of p primes is
odd if p is odd and even if p is even. I came to this conclusion as
follows:

Assuming the goldbach conjecture, then an even number is the sum of two
primes.
Then an even number + 1 = prime + prime + 1.
Or, equivalently, an odd number = prime + (prime + 1).
But, given that the prime is not two, then (prime + 1) is an even
number.
And, an even number is the sum of two primes, so, the sum of three
primes is an odd number, if and only if none of the primes are two.
Then, an odd number + 1 = prime + prime + (prime + 1) and we can start
over using the same method.

Can anyone see a problem with this or does anyone know if this has
already been documented? I figure that this is reasonably significant
if it is correct.

Thanks for your time,

Lynton

.

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