What are some equivalent statements of (strong) Goldbach Conjecture?

Question

What are some equivalent statements of (strong) Goldbach Conjecture ?

We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens function, error terms of Prime Number Theorem, and Farey sequences. Those equivalent statements do not use Riemann Zeta function directly, but provide additional insights about Riemann Hypothesis from very different angles.

What are some equivalent statements of (strong) Goldbach Conjecture ? to shed lights on this problems from different angles ?

Answer 1

For every integer $n \geq 1$ there exists primes $p$ and $q$ such that $\varphi(p)+\varphi(q)=2n$

where $\varphi$ is Euler's Totient function .

Answer 2

For every integer $n \geq 2$ there exist integers $k, p$ and $q$ with $0 \leq k \leq n-2$ and with $p$ and $q$ prime such that $n^2 - k^2 = pq$. (http://www.maa.org/sites/default/files/Reformulation-Gerstein20557.pdf)

Answer 3

"Every natural number greater than one can be written as the arithmetical mean of two primes (not necessarily different)"

... from my point of view, it simplifies the restrictions (to be "greater than one" is simpler than to be "even and greater than two"); and it is intuitively better, as "arithmetic mean" gives a constructive recipe to obtain the number as the center of their associated "prime couples"

Answer 4

$$\sum_{_{3\leq p\leq2n-3}}\omega\left(2n-p\right)>\sum_{_{3\leq p\leq2n-3}}\pi_{p,b}\left(2n-3p\right)$$

where $p$ is prime, $\omega$ is the prime omega function, $\pi_{a,b}$ is the modular prime counting function and $b=\left(2n-3p\right)\mod p$.