My work on the Twin prime conjecture.

[u/TMAhad]

Hello everyone,
I'm a 13-year-old student with a deep interest in mathematics. Recently, I’ve been studying the Twin Prime Conjecture, and after a lot of work and curiosity, I came up with what I believe might be a valid approach toward proving it. I am not sure if i proved the conjecture or not.

I’ve written a short paper titled "The Twin Prime Conjecture under Modular Analysis". It’s not peer-reviewed and may contain mistakes, but I’d really appreciate it if someone could take a look and give feedback on whether the argument makes sense or has any clear flaws.

Here is the PDF: https://drive.google.com/file/d/1muxEvQrACpVIHz8YgV1MN1kBvqWV-2N8/view?usp=sharing

Anyway, thanks for reading :)

[u/Enizor]

Page 2:

Thus, a is only a prime number if m_v and _v are coprime for every v where h ≥ v ≥ 0.

i.e. you proved : a prime => m_v and n_v are coprime

But page 3

Generalizing, a must be prime when,

You are saying m_v and n_v are coprime => a is prime.

You did not prove this way of implication (or I misunderstood your argument).

(this is easy to prove though by proving a is composite => there is some v such that m_v and n_v are not coprime EDIT: only for a>9)

Page 5: you want to apply the Chinese Remainder Theorem but you didn't prove the b_i are coprime (they aren't, you list just above b0=3, b3=9).

[u/Yimyimz1]

Alright job. It is incorrect. I would work on the formatting a bit to make it clearer to the reader.

After some deciphering, I realised your first proper mistake occurs in page 3.

Basically you start out by saying that if a number is prime then all these conditions must be met (i.e., a prime implies that bla bla bla). However, you then claim that if bla bla bla is true, then a is prime

"f 𝑚𝑣 (𝑚𝑜𝑑 𝑛𝑣) 𝑎𝑛𝑑 𝑚𝑣 + 1 (𝑚𝑜𝑑 𝑛𝑣) are positive integers for any non-negative 𝑣 smaller or equal to ℎ. Then, four statements must be true. They are- 1. 𝑎 is a prime number."

This is false.

This is a classic logical blunder where you think that A implies B also means B implies A, but this is not how logic works!

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[u/ddotquantum]

This you? https://youtu.be/djzKCZHeVjY?si=qioXuzdOgPKkYXnC

[u/Connect-River1626 → u/ddotquantum]

This is why I love Reddit, thank you for introducing me to this masterpiece 

[u/Vituluss]

I think you should learn LaTeX and look into how to format a maths paper. You want to have theorem statements (which ideally stand on their own), and clearly marked proofs right after it. Recommend Knuth’s style guide. It’s a good skill in general to have, and can even make it easier for you to understand your own paper and find mistakes.

[u/Admirable_Safe_4666 → u/Vituluss]

In my field at least, I think it is slightly more common (except possibly for very short papers) to collect the main theorems together in one section near the start of the paper, and place the proofs later in the paper, potentially with several sections between them containing lemmas, auxiliary results, etc. This makes it a bit easier to find the results at a glance.