In 2013, when I was just a totally newbie recreational mathematician, I read about Levy's conjecture (i.e., Lemoine's conjecture, stating that all odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime) and I came up with the idea that, by assuming Goldbach's strong conjecture as true, we would have easily deduced the following two conjectures (or at least one of them).
Conjecture 1: For every prime number $p_0 \geq 7$, there exists (at least) one pair of distinct primes ($p_1, p_2$) such that $p_0=2 \cdot p_1+p_2$.
Conjecture 2: $\forall n : n \in \mathbb{N}-\{0,1,2,3,4,5,6,7\}$, there exists (at least) a couple of odd primes, $p_1 \neq p_2$, such that $2 \cdot n+1=2 \cdot p_1+p_2$.
About Conjecture 1, the number of ways such that $p_0=2 \cdot p_1+p_2$ seems to increase almost linearly (see Figure below) and a brute force test has been performed up to $746562601=2 \cdot 7+746562587$, confirming the statement for every $p_0 \leq 746562601$.
Number of ways to write $p_0$ as $2 \cdot p_1+p_2$
Any chances to prove the above by taking the strong Goldbach Conjecture as true?
Best Answer
This is an expansion of the comment above and a partial answer.
A theorem of Daniel Shiu might be partially useful in proving Conjecture 1.
We use $P_1, P_2$ to disambiguate the notation between $k$-th prime denoted by $p_k$ and $p_1, p_2$ used in the OP's notation.
Using the theorem, if we set $a = P_2, q = P_1$ we will obtain an infinite set of primes satisfying Conjecture 1 with $P_0 = 2P_1 + P_2$.
What remains to be proved is that this covers all prime pairs satisfying Conjecture 1.
References:
D. K. L. Shiu, Strings of Congruent Primes, Journal of the London Mathematical Society, Volume 61, Issue 2, April 2000, Pages 359–373, https://doi.org/10.1112/S0024610799007863