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