Dtyjlui

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 ?

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)

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 .

"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"

Reformulating the answer of @LorenoHeer, using the fact that the sum of the first $n$ odd numbers is equal to $n^2$: Every term $(2n-1), nge4$ of the sequence of odd numbers is the final term of at least one series containing a prime number $p$ consecutive terms, the middle term of which is a prime number $q$.

It can be seen that the sum of such a series is $pq$, that is, the number of terms $(p)$ times the average value of the terms $(q)$. The series is also the sum of the first $n$ odd numbers $(=n^2)$ less the sum of the first $k$ odd numbers $(=k^2)$, or $n^2-k^2$, as given in the referenced answer. Since $(2n-1)$ is the $n^{th}$ odd number and there are $p$ terms in the series, $k=n-p$.

It is unsurprising that one can pick an odd prime and make it the center term in a series of consecutive odd numbers with a prime number of terms. What strikes me is that if one does so systematically, the set of final terms of all such series saturates the (suitably large) odd numbers. Tangentially, this says something about the distribution of prime numbers; whether what it says is distinct from what the canonical formulation of the Goldbach conjecture says about the distribution of prime numbers is beyond my ken.

Here is a more abstruse equivalent to the Goldbach conjecture. Consider (sufficiently large) even numbers $2m$. If $m$ is prime, the conjecture is true, so we omit that possibility from further consideration. For odd prime numbers $3leq p_i<m<p_j<2m$, let $n=prod{p_j}$. Let $c_i=2m-p_i$, and $a=prod{c_i}$. Then $left(frac{a}{n}right)=0$ where $left(frac{a}{n}right)$ is the Jacobi symbol.

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!