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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 $p_0 \leq 746562601$.

Ways to write p_0 as 2 \cdot p_1+p_2 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?

vvg
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Marco Ripà
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    Lemoine's Conjecture: $2n + 1 = p + 2q$ always has a solution in primes $p$ and $q$ (not necessarily distinct) for $n \gt 2$. Conjecture 2 above is the same as Lemoine's with the removal of the not necessarily distinct qualifier and the lower bound for $n$ raised to $8$ from $2$. That sounds like there might be a counterexample. – vvg Dec 02 '22 at 03:15
  • @vvg Yes, I think you are right. Anyway, would it be possible to keep the additional constraint, $p_1 \neq p_2$, if we are taking the strong G.C. as true by hypothesis? – Marco Ripà Dec 02 '22 at 03:30
  • There is a theorem of Daniel Shiu about strings of primes congruent to an integer. I had posted a question on it here: https://math.stackexchange.com/questions/4545627/obtaining-strings-of-congruent-primes-shius-theorem It appears that Shiu's theorem may be partially useful. For eg: In the string of congreunt primes $p_{n+1}, p_{n+2}, \dots p_{n+k}$ such that $p_{n+1} \equiv p_{n+2} \equiv \dots \equiv p_{n+k} \equiv a \pmod{q}$, we can set $a = P_2, q = P_1$ to obtain an infinite set of primes satisfying Conjecture 1. Proving it covers all prime pairs satisfying Conjecture 1 will remain. – vvg Dec 02 '22 at 03:42
  • Very interesting, I think this comment is worth to be posted as an aswer (so I can upvote it too). – Marco Ripà Dec 02 '22 at 03:45
  • How far did you check conjecture $2$ ? – Peter Oct 12 '23 at 10:52
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    At the time, we verified that Conjecture 2 holds as long as $2 \cdot n+1 \leq 746562601$. – Marco Ripà Oct 12 '23 at 12:00

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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.

Theorem (Shiu, 2000): Let $p_n$ denote the $n$-th prime. $\forall k \in \mathbb{N}$ and $a, q \in \mathbb{N}, q \ge 3$ having $\gcd(a,q) = 1$ there exist ‘strings’ of congruent primes $p_{n+1}, p_{n+2}, \dots p_{n+k}$ such that $p_{n+1} \equiv p_{n+2} \equiv \dots \equiv p_{n+k} \equiv a \pmod{q}.$

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

vvg
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