Any even $n> 10$ can be written as $n = pp'+p''$ for some primes $p,p',p''$

Question

Conjecture:

Any even integer $n>10$ can be written as $n=p\cdot p'+p''$, for some $p,p',p''\in\mathbb P$.

Verified for all $n<1,000,000$.

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I first intended to post the weaker conjecture: $\mathbb P\subset\Big(\mathbb P\cdot\mathbb P+\mathbb P+1\Big)\cup\{2,3,5,11\}$.

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Answer 1

I wouldn't be suprised if this is extremely hard to prove and currently open.

Chen's theorem implies that every sufficiently large even number can be written as $n=q\cdot q'+q''$, for some $q',q''\in\mathbb P$ and $q' \in\mathbb P \cup \{1\}$.

In 2015, an explicit bound was given for sufficiently large: $n>e^{e^{36}}$ is enough.

However, Chen's theorem is weaker since $q'$ might be $1$ instead of a prime number.