I have tried searching for the following and I mostly got results dealing with: the rarity of different gaps, twin gaps, cousin gaps...
As a self learner, there is a chance that I didn't know of a better way to search, So if this is a duplicate please close and refer me to it.
I doubt that there is any proof, but I would deeply appreciate any references or any heuristics or any counter example.
The question is quite simple:
Does any odd prime number belong to at least one set of a minimum of $3$ primes numbers, that are separated from each other by the same gap?
*Note: The prime numbers themselves aren't necessarily consecutive in nature.
I am aware that within large prime numbers there are different behaviors, but there are also many more options.
Here are some basic examples:
$3 , 5 , 7$ gap: $2$
$5 , 11 , 17 , 23, 29$ gap: $6$
$7 , 13 , 19$ gap: $6$
$19 , 31 , 43$ gap: $12$
$23 , 41 , 59$ gap: $18$
$23 , 47 , 71$ gap: $40$
$3 , 53 , 103$. gap: $50$
$79 , 109, 139$ gap: $30$
...