I am currently working quite hard and putting in a decent amount of effort in to my first mathematical thesis. As a high school student, this is becoming increasingly difficult. Because of my fear of having my ideas stolen, I kindly will refuse to state the abstract of that thesis. I am writing it for two reasons: 1) It would look like gold on college apps and 2) I love the type of math I am focused on.

My question (sorry for the fluff): Can I have a proof or disproof that there exist finitely many $P_{2n-1}$ primes that are expressible as a $3n-1$?

Note:

• $P_{2n-1}$ primes denotes every other prime beginning with the first prime. This would be an example of that set of primes: $2,5,11,\cdots$. Oeis sequence

• Primes represented as $3n-1$ are examples of the primes such that $3n-1=p$ has solutions. Oeis sequence

At least with an answer to your question, my thesis can go one way or another, but as this is really computational to generate a given quantity, I should get a solution with any answer I get. That means that this statement could be incorrect.

Key point- I have 4 major roadblocks here to creating my thesis, and out of respect for the mathematical community, I will solve the other 3 on my own. This is the highest priority road block though.

I have no clue how to prove this question whatsoever.