Goldbach Conjecture predicate form?

Question

I am learning logic, and when I was taking a quiz one of the multiple choice questions was "Which of the following is an unsolved conjecture?" I picked the following answer because I thought it was the Goldbach Conjecture. However, this is not the right answer. What am i missing?

$\forall m \in \mathbb{N}, \exists n \geqslant m, \text{ $n$ even},\exists p,q \in P, n=p+q$

The correct answer was: $\forall m \in \mathbb{N}, \exists n \geqslant m, n\in P \text{ and } n+2 \in P$

However, why Is the first one not also correct?

Answer 1

It looks like the "correct answer" is a statement of the Twin primes conjecture.

Answer 2

The statement $$ \forall m \in \mathbb{N}, \exists n \geqslant m, \text{ $n$ even},\exists p,q \in P, n=p+q $$ means that for any $m$, THERE EXISTS a larger-than-$m$, even integer that is the sum of two primes. This is obvious--just take two larger-than-$m$ primes and add them together.

The Goldbach conjecture instead says that ALL even integers ($\ge 4$) are the sum of two primes, not just that THERE EXISTS sufficiently large such even integers. It's a much stronger statement.