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Twin Primes… (Revolution Way )

20 July 2026, Version 2
Working Paper
This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.

Abstract

This work proposes a structural approach to identifying infinitely many twin prime pairs by examining arithmetic progressions generated from the linear forms y3=3+2x and y5=5+2x. By analyzing values of x that simultaneously produce primes in both sequences, the method partitions the natural numbers into excluded subsets—specific odd and even values of x that yield composite outputs—and an infinite admissible subset A for which both expressions generate prime numbers. Since the excluded subsets are infinite and proper subsets of the positive integers, the remaining admissible set A is also infinite. Each x∈A corresponds to a pair (y3,y5) differing by 2, forming a twin prime pair. Therefore, the structure of the admissible set implies the existence of infinitely many twin primes. This framework is presented as a conceptual challenge to the long standing stagnation surrounding the twin prime problem.

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Comment number 10, Taha Muhammad: Jul 23, 2026, 13:07

Your words are unnatural, and there is no need for me to interfere in your personal actions. You are free to do and say whatever you want, and thank God I have millions of people of truth who notice what you write.

Response,
Peter M :
Jul 23, 2026, 14:08

You say I am free to say whatever I want. That means that I am free to comment on each of your papers and point out their many basic flaws and mistakes. Whether you actually engage with the feedback (or continue to ignore it) is your decision.

Comment number 9, Taha Muhammad: Jul 23, 2026, 02:42

1) You did not answer my Q: you are unfortunately, not authorized to endorse anyone for arXiv! 2) I solved Collatz in 5 ways, Fermat & its General Case in 8 ways, Euler only one-way, Cubic root by hand and any for any real number. If you are able to make arrangement on stage with math judges on TV between you and I, then the world will see the fact. I like math talk.

Response,
Peter M :
Jul 23, 2026, 03:04

I am an endorsed author on arXiV, but in a different field of mathematics, and cannot endorse number theory papers. Either way, you are not getting an endorsement on arXiV. No legitimate researcher is going to endorse abysmal work such as yours. You have not solved any open problems, and saying that you have does not make your claims true. Behave like an adult or otherwise I will report your content and request it be removed.

Comment number 8, Taha Muhammad: Jul 22, 2026, 16:26

1) Mr. Peter M., you are unfortunately, not authorized to endorse anyone for arXiv. I was merely being polite. Your attack on me has no place in scientific discussion. 2) Thank you, Cambridge Open Engage for letting my solutions online for entire world. 3) Thank you, UK, for having great Cambridge University to help knowledge of entire world.

Response,
Peter M :
Jul 22, 2026, 17:16

No, you are simply refusing to acknowledge any possibility that you are incorrect and that you have not proven what you think you've proven. That is the literal definition of a crank, so I am not attacking you. I am trying to help you realize your errors but you are refusing any and all criticism. I don't think you should even be posting on Cambridge Open Engage, and even if you do, it will never be accepted as submissions here are not peer-reviewed.

Comment number 7, Peter M: Jul 22, 2026, 03:02

No, you have not solved the Collatz conjecture, Euler's box problem or any other open problem you claim to have solved. You are what the math community calls a "crank". You have not proven that there are infinitely many pairs in TP (e.g., one for each set), and if the proof was that obvious it would have been solved centuries ago. I am 100% certain that you do not understand mathematical research and that is why I will not endorse you on arXiV. You need to get a grip on yourself and stop wasting your own time if you want what's best for yourself.

Comment number 6, Taha Muhammad: Jul 22, 2026, 01:11

Ans: What is [x in N_{pn} ]? I have infinite subsets of N+!

Response,
Peter M :
Jul 22, 2026, 01:33

The fact that you have infinitely many subsets of N+ is irrelevant. You would need to somehow prove that for each of these sets (N_{7n}, N_{11n}, N_{13n} etc.), there exists an x that generates a twin prime pair. This is already at least as hard as proving the twin prime conjecture itself, and you have not shown this. Sorry, but you are completely and utterly clueless on how proofs work, and how prime numbers are structured. I guarantee you that if you were to submit this to a top number theory journal, it would be immediately rejected.

Response,
Taha Muhammad :
Jul 22, 2026, 02:42

You are right, but my job is to discover and to organize infinity sets for world and at least in each one infinity set it will be an element will get pair of TP. Very clear that I have infinity pairs of TP members. Thus, TP is infinite set. Dear professor, I really appreciate your time for this nice of your ideas and I am your student, but I solved Collatz, Euler, Fermat, and many ...Then where is your endorsement to my discoveries to arXiv .

Comment number 5, Taha Muhammad: Jul 21, 2026, 21:01

1) If you look at this: The above processes will continue forever to infinity to create infinite sets, such that each element of (some odd and some even) values (x) provides one element of TP 2) If in each set from infinity sets at least one x value provides one pair of TP, then the total pairs will be infinity.

Response,
Peter M :
Jul 21, 2026, 21:41

And how do you know that for each set N_{pn} = {p, 2p, 3p, ...} where p is not 3 or 5, you can find some x in N_{pn} that (p,q) = (2x+3,2x+5) is a twin prime? You do not prove this at all. All you have shown is some semblance of a claim that is pretty much impossible to verify for p not equal to 3 or 5.

Comment number 4, Taha Muhammad: Jul 21, 2026, 12:05

Yes, dear professor, you are right and I told too that I have to wait millions of years to find next pair of set TP, but there is no end for pairs of TP! This is correct: "Yitang Zhang's 2014 paper that shows that there are infinitely many prime pairs that differ by 7 x 10^7". always by adding 2s --- > pair of odds be TP or Not, if not continue adding 2s to get pair of TP under control of logic (or). If you reach final pair of TP, it means you are stopping adding 2s.

Response,
Peter M :
Jul 21, 2026, 14:32

This is completely incorrect. First, if a proof was that simple, why has the problem remained unsolved? Second, you need to prove that a larger twin prime pair always exists; simply adding 2s doesn't cut it. What if I claimed that there were infinitely many primes p such that p, p+2, and p+4 were prime? Could I use your exact same argument and keep adding 2 until I obtain the next triple of primes? The answer is absolutely not.

Comment number 3, Taha Muhammad: Jul 21, 2026, 04:51

1) The answer is: N+ infinite/N even infinite = N odd infinite too. A = N+ infinite / (B u C) infinite = A infinite too. 2) Read Taha's Infinity fact: 1 cm segment has infinity points = infinity zeros, 2 cm segment has infinity points = infinity zeroes, 100 km segment has infinity zeroes. 3) Prime numbers are less frequent as you have larger numbers ...Yes, but we are not talking about density, it is adding 2s until you get both (p,q) in TP. Taha prime detector showed that., and infinity sets, with infinity elements will have x value to make both y3, y5 primes ate the same time, but I have sometimes to wait millions of years until I find correct x to make p, q primes. Thank you. My Solution is Correct.

Response,
Peter M :
Jul 21, 2026, 05:38

No, your solution is completely incorrect. I know what I am talking about. I regularly teach courses on proofs and number theory, and your solution does not prove anything. Simply saying that there are infinitely many x's and eventually you'll find the next x such that 2x+3 and 2x+5 is not a rigorous argument. How do you know such an x exists, for all x > N? You do not, and no one does. Please read Yitang Zhang's 2014 paper that shows that there are infinitely many prime pairs that differ by 7 x 10^7. It was published in the Annals of Mathematics, a highly prestigious journal. https://annals.math.princeton.edu/wp-content/uploads/annals-v179-n3-p07-p.pdf

Comment number 2, Taha Muhammad: Jul 21, 2026, 01:14

Peter M, Jul 20, 2026, 10:10: 1)Taha, your updated paper still contains extremely basic errors and does not address any feedback I have given you. 2) Yes, B u C is infinite. But you have no reason to claim that A = N+ / (B u C), where / is the set difference operator, is infinite. It could be either finite or infinite. Taha's Answer: Answers to honorable "Peter M": 1) I did not agree with your feedback on my version 1. This solution (version 2 has same idea) with new steps. 2) N+ infinite/ N even infinite = N odd infinite too! This one below is the same above: A = [N+ / (B u C)] infinite too. Infinite set N+/ infinite set N even = infinite set N odd. Thank you for your time.

Response,
Peter M :
Jul 21, 2026, 03:38

You are not understanding any of my feedback. It is obvious that B and C are infinite sets. For example, B contains all odd multiples of 3 and C contains all even multiples of 3. The union (B u C) is also infinite. But you have no reasoning why N+ / (B u C) is also infinite. What if B u C contains all natural numbers greater than some constant? Prime numbers are less frequent as you have larger numbers - this is a well known fact in number theory. You should not be attempting these problems as it's abundantly clear you are unable to work with mathematical proofs. Not to be blunt, but I've seen better proofs from undergraduate students. Continuing on these problems will only waste your time, as they will never be accepted by the mathematical community.

Comment number 1, Peter M: Jul 20, 2026, 15:10

Taha, your updated paper still contains extremely basic errors and does not address any feedback I have given you. Yes, B u C is infinite. But you have no reason to claim that A = N+ / (B u C), where / is the set difference operator, is infinite. It could be either finite or infinite.

Version History

Jul 20, 2026 Version 2

Version Notes

I made it very simple

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License

All Rights Reserved

DOI

10.33774/coe-2026-vp5c4-v2
D O I: 10.33774/coe-2026-vp5c4-v2 [opens in a new tab]

Author’s competing interest statement

The author(s) have declared they have no conflict of interest with regard to this content

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