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

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 ?

Solution 1:

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)

Solution 2:

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 .

Solution 3:

I conjectured this:

Every integer $n>3$ is halfway between $2$ primes.

See the proof here .

By the way, this is a very good question. Like you said, the best way to solve this old problem must certainly be to look at it from new angles. There should be a lot more answers here. I have 2-3 other statements in mind. I'll do some searching and post them soon!

Solution 4:

"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"