Number Theory

   

Infinitely Many Twin Prime Reexplored - Proof

Authors: James DeCoste

Without any concrete feedback from the mathematical community, I was forced to have an enlightening discussion with Google's AI engine to determine what the likely problems were with my initial paper Infinitely Many Twin Primes — Proof (found on viXra.org: https://vixra.org/abs/2502.0186).Since I did all my own research in a sandbox without contributions from the mainstream mathematical community ( blind research ), I evidently ran into the same wall all prior mathematicians hit and that was proving that the twin primes candidates don't simply fizzle out as one approaches infinity. I was relying on combinatorics to prove this, which I now see was a mistake when taken by itself. As the AI engine and myself did a deeper dive on current mainstream research we noticed that much of the research I performed but did not include in my original paper were already explored by others. Some of these will be the key when reworking this on it's second take; the Hardy-Littlewood ratios 1:1:2 and the product rule that sees the pattern repeat at that product, to mention two. This was done blind without ever realizing it.We were able to determine that the rest of the original paper was sound even if not written in mainstream mathematical language. However that wall where the twin primes could simply fizzle out or completely dry up was impossible to ingore. While exploring some of the new approaches being implemented by Maynard and Tao, I made the realization that the probabilities angle should not have been abandoned. I had already done research to establish floor and ceiling limits as decaying log curves...but they too appear likely to fizzle out as one approaches infinity. That was until we took a look at a specific probability ratio of my window size sample growth versus the number of primes total growth. The first was a quadratic growth rate ( x^2) and the second was strictly linear ( number of primes grow at a steady rate). Using this probabilty along with the prior published paper finally plugs the hole and smashes down that wall as a mathematical improbability. The idea is that we have a fraction that is continually shrinking but can't actually get to absolute zero, the wall. The number of new elements (prime elimination patterns) is not increasing fast enough to keep up and eliminate all candidates in the exponentially increasing window size of my ranges. Candidates always escape. This clearly shows that my combinatoric approach was also correct in it's logic and that twin primes can never totally disappear.

Comments: 12 Pages.

Download: PDF

Submission history

[v1] 2026-05-27 22:09:00

Unique-IP document downloads: 8 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.

comments powered by Disqus