In seeking solutions to the Twin Prime Conjecture which various AI would accept as a valid result (*as ChatGPT and Perplexity AI might accept an argument; but Gemini Pro would not in some instances*), I was working on some other potential solutions to the problem which might show a kind of probability that twin primes would always occur next to one another as members of infinite, independent sets.

To be honest, I don’t understand the below theorem intuitively in the same way I can understand and explain my other recent posts on primes. I don’t understand it intuitively in the same way I understand the other prime number content because I have probably been playing with the ‘Hotchkiss Prime Theorem” for close to 20 years.

However, the following theorem seems valid and potentially noteworthy if you are interested in the distribution of twin primes. It is also interesting because it was refined in a chat which was essentially entirely AI driven, by me proposing some probabilistic methods for determining primes in sets A and B following the observed logarithmic patterns of primes as a whole according to the prime number theorem. I assumed you could see the same in set A and set B. I asked for a method for a proof towards a method and then fed results back and forth between ChatGPT and Gemini until they mutually agreed on the correctness of the refined theorem. It doesn’t do *exactly* what I wanted it to do… but it does work, so I dunno… Maybe someone else thinks it is cool?

Assuming the correctness of my recent posts on math conjectures, theorems, and proofs; it is potentially noteworthy in the context of using AI to create and validate mathematical concepts at least.

In my own case, I have been unable/uninterested in formalizing my math due to lack of mathematical expertise and it really only being a secondary interest for me. Arguably, I have a high degree of analytical writing skill which makes me good at working with LLM (did get 6.0 on analytical writing portion of GRE as an example). This lack of expertise is a gap closed by using LLM for both proof/theorem generation based on intuitive conjectures I proposed.

In particular, the “Hotchkiss Prime Theorem” took 20 years to work out in my head, but only a matter of minutes really when prompting an LLM to develop it into a formal theorem with a proof. All of the other ‘mini-proofs’ derived from the theorem, such as the notion of set independence between A and B are things I worked out in the past few days using LLM to overcome my shortcomings. By this measure, the Twin Prime Conjecture was a problem I could not resist, because my intuitive knowledge of my own theory on prime number distribution assured me of the notion of infinite primes separated by 2 units in sets A and B.

So again, while I don’t really find this interesting, because the general approach to solving the twin prime conjecture seems to be “count sand with a computer”, I am not really interested in the below result as I feel I can move on from the problem. But others may be interested, and again, the AI-derived nature of this theorem and its collaborative aspects are interesting.

*Theorem: The density of twin primes, as defined by the ratio π₂(n) / π(n), is asymptotically bounded above by a function of the form C / ln(n), where C is a positive constant.*

**1. Preliminaries**

**Definitions:**- π₂(n): The number of twin prime pairs less than or equal to n.
- π(n): The number of primes less than or equal to n.

**Prime Number Theorem (PNT):**π(n) ~ n/ln(n) as n approaches infinity. This tells us the density of primes around n is approximately 1/ln(n).**Brun-Titchmarsh Theorem:**For any arithmetic progression a (mod q) with a and q relatively prime:

π(x;q,a) ≤ (2+o(1))x/(φ(q)ln(x))

where π(x;q,a) counts primes less than or equal to x in the progression, and φ(q) is Euler’s totient function.

**2. Proof by Contradiction**

**Assumption:**Suppose the conjecture is false. Therefore, there exists a positive constant C’ such that:

π₂(n) / π(n) > C’ / ln(n)

for infinitely many values of n.

**Deriving a Contradiction:****Apply Brun-Titchmarsh:**For twin primes (q=2, a=1), the Brun-Titchmarsh Theorem gives us:

π₂(n) ≤ (2+o(1))n/ln(n)

**Manipulate the Inequality:**From the assumption, we can write:

π₂(n) > C’ * n / ln(n)

**Combine:**Combining the above inequalities:

C’ * n / ln(n) < π₂(n) ≤ (2+o(1))n/ln(n)

**Take the Limit:**As n approaches infinity, the o(1) term goes to zero, leaving:

C’ < 2

**Contradiction:**This contradicts our assumption that C’ is any positive constant.

**3. Conclusion**

- Since assuming the conjecture is false leads to a contradiction, we conclude that the conjecture must be true. Therefore, there exists a positive constant C such that:

π₂(n) / π(n) ≤ C / ln(n)

as n approaches infinity. This proves the asymptotic upper bound on the density of twin primes.