Is there the following generalization of Goldbach's weak conjecture?

Question

I heard that the following has been proved: Every odd number greater than 7 can be expressed as the sum of three odd primes. What do we know about the following?

There is a $k\in\mathbb N$ such that for every $n\in \mathbb N$, $2n+1>7$ there exists primes $p_1,p_2,p_3$ such that $2n+1=p_1+p_2+p_3$ and $p_1<k$.

I mean, can we say that we can always represent all odd numbers greater than $7$ as a sum of three primes where the smallest prime is bounded by some constant.

Answer 1

Define a Goldbach number to be an even integer that is the sum of two primes. Note that your statement is equivalent to the following: the gaps between consecutive Goldbach numbers is at most $k$ (perhaps even $k-3$). This is still an open problem, as mentioned in this Terry Tao blog post. It is known (as you can read about here) that you can replace the constant $k$ with a power of $\log n$ and the statement becomes true, which is close at least.