Assuming the Even Goldbach conjecture is true, it will mean that every even number greater than $2$ as $E$ can be represented as the sum of $2$ primes as $p$'s
Assuming the Odd Goldbach conjecture is true, it will mean that every odd number greater than $7$ as $O$ can be represented as the sum of $3$ primes as $p$'s
Combining all the possibilities, will all $O$ 's and $O ⋅ 2$ 's share at least $1$ of the $p$'s in such a way that the remaining are also $p$'s ? Will there be a way to prove it?
Examples:
$7 = 3 + 2 + 2$ and $14 = 3 + 11$ ($3$ in both)
$9 = 2 + 2 + 5$ and $18 = 13 + 5$ ($5$ in both)
$11 = 5 + 3 + 3$ and $22 = 3 + 19$ ($3$ in both)
$13 = 7 + 3 + 3$ and $26 = 7 + 19$ ($7$ in both)
$15 = 2 + 2 + 11$ and $30 = 11 + 19$ ($11$ in both)
$17 = 7 + 5 + 5$ and $34 = 5 + 29$ ($5$ in both)
$19 = 7 + 7 + 5$ and $38 = 31 + 7$ ($7$ in both)
$21 = 11 + 5 + 5$ and $42 = 11 + 31$ ($11$ in both)
$23 = 5 + 5 + 13$ and $46 = 41 + 5$ ($5$ in both)
$29 = 5 + 5 + 19$ and $58 = 53 + 5$ ($5$ in both)