Let $\beta(n)$ represent the number of Goldbach prime pairs that each add up to an even integer $n$.
Observation: If $p$ is a prime, for $n \ge 152$, (ignoring $n$ = powers of $2$)
$$\beta(n) \le \beta(p*n) < p*\beta(n)$$
For powers of $2$, the above inequality is true for $n \ge 128$.
Since any number $m$ can be obtained by multiplying primes, replacing $p$ with $m$ in the above inequality would work as well.
Implication: If we start with some $n = 2p$, we know that $\beta(2p) \ge 1$. And so proving the above inequality would prove GC.
Question: If this inequality is indeed true, how would one go about proving it? What approaches would you take? Any hints? Thanks
68 can only be written in two ways as the sum of 2 primes (7 + 61, 31 + 37), however 34 can be written in 4 ways as the sum of 2 primes (17 + 17, 31 + 3, 23 + 11, 29 + 5).
Then $\beta(68) = 2$ and $\beta(34) = 4$ and $68 = 2 * 34$, with 2 being prime...
I am definitely not sure that your inequality is correct. We also have $\beta(152)=4$ and $\beta(76)=5$.