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Twin Primes… (Revolution Way )

11 July 2026, Version 1
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This content is an early or alternative research output and has not been peer-reviewed by Cambridge University Press at the time of posting.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 8, Taha Muhammad: Jul 16, 2026, 16:28

Really, did you see my Collatz solution in 4 ways, and Euler Perfect Box? Please, just look at them if you have a time. Thank you, anyway.

Response,
Peter M :
Jul 16, 2026, 17:29

Yes, and those also have extremely basic logical and mathematical errors that make the proofs wrong. I would rather comment on those separately, since this submission is for your twin primes claim.

Comment number 7, Taha Muhammad: Jul 16, 2026, 15:47

No please, I am not hiding myself. It is nice talk to you, if you ask me, then I will answer, but my level is 6 to 7 grade student + thinking + not giving up + trying + I need your help please. Do you know I solved Collatz and Euler, and many ...? I have master's degree in math from UND University 1995 with not enough English, but proud of my Civilian USA Army ended 2010, and I am teacher (not now, no job, my pension $274/month) since 1967 Kurdistan of Iraq. My books at Amazon.com "We Survived Iraq and Turkey...Long Road to Freedom. Can you please indorse me for arXiv and I have no one to tell accept you! That is why I need to win some $s to get pay for my mortgaged house.

Response,
Peter M :
Jul 16, 2026, 15:59

I cannot and will not endorse you on arXiv, and I won't because I am absolutely certain that your papers are far below the standards expected in mathematical research. I have a Ph.D. myself, and I have taught undergraduate-level number theory and proof-based courses. Your papers would not even qualify for a passable homework assignment in one of my classes, to be honest. You need to come to grips with the fact that you have not solved the problems you claim to have solved. If your goal is to win the prizes associated with the Collatz conjecture or other difficult problems (e.g., Millennium problems), I can guarantee that will not happen in a million years with the papers you are posting here.

Comment number 6, Taha Muhammad: Jul 16, 2026, 14:31

Your opinion differs from mine, yet we do not differ on a human level. At the age of 82, I am attempting to resolve what others could not. There is no need for us to discuss this any further.

Response,
Peter M :
Jul 16, 2026, 15:17

I am trying to help you recognize your errors, but you are simply dodging any and all criticism of your work. If you are trying to get your work recognized by the mathematical community, that will never happen as this falls way below the standard of mathematical proof; moreover, it is simply wrong. It is not because you do not have a Ph.D. or claim to be an independent researcher. By refusing to discuss further, you are only hindering yourself.

Comment number 5, Taha Muhammad: Jul 16, 2026, 06:43

Due to the lack of fair competition and the presence of judges who are mathematicians, the independent researcher remains overlooked. My view—that I have achieved a solution no one else has—should not be met with a harsh response from anyone. The proper response is to compete on television before judges. My statements are entirely professional, and I am proud of my solutions, especially the one regarding the Collatz conjecture. There is no need for condescension or arrogance; such attitudes are frowned upon in society. My solutions are at Cambridge, and you are free to think whatever you wish. Thank you for your opinion.

Response,
Peter M :
Jul 16, 2026, 07:40

We are free to critique your work as harshly as possible without belittling others. I feel you are afraid to face the fact that you have not proven anything. Many people besides you claim to have solved the Collatz conjecture or the twin prime conjecture, etc. These are extremely dangerous problems to attempt because of their simple problem statement, but are in fact extremely difficult problems. A 1-page "proof" with elementary methods is not going to cut it. You need to do your research, as it is obvious you haven't.

Comment number 4, Taha Muhammad: Jul 16, 2026, 03:52

You: 1. x is not a multiple of 11. When x=88, (179,181) is a twin prime pair. Taha: [[[Yes, “you are right”, and I am wrong because I had no time to check all 11n, n in N+, but some 11n are not values for x. Can you check 11n when n>5^7 creates TP or not? [ this rush mistake by me is not related to my level “low or high”! Then so what you are PhD or Einstein and did not solve Collatz, Euler, Fermat in one page, any root by hand for any real #, Taha’s Geometry Theorem, and many …!]]] You: 2. The fact that A is a subset of an infinite set does not imply that A is an infinite set, because finite sets are also subsets of an infinite set. Taha” [[[ N+ infinite/ N even infinite= N odd infinite]]] (*) Since A elements are many and the graph elements goes to infinity, the use (*)! You: 3. The statement A <--> TP is meaningless because A and TP are sets and not true/false statements. Do you mean to say A is the set of twin primes? Or the set of values x such that (3+2x,5+2x) is in TP? Taha: Each x in A provides one element of TP, and then A infinite will provide TB infinite. You: Taha: Either way, you have no argument that TP is infinite. Taha: Answer above. You: elementary methods like yours almost certainly won't resolve it. Taha: You and entire mathematicians did not each of the above in one page. Its not shame I solve problems and have no PhD! U R talking about my level and I solved Collatz in 5 ways, Euler Perfect Box, and many, you and I should tell all levels are ok and respected.

Response,
Peter M :
Jul 16, 2026, 04:26

That graph showing y=2x+3 and y=2x+5 does not show anything about the structure of prime or twin primes. Your set N+ / (some N_odd U some N_even) is not well-defined. Even then, if A = N+ / {some infinite set}; that does not indicate at all whether A is infinite or not. For example, A could be N+ / {all natural numbers greater than 10^100}, then A is not an infinite set. If this is the research you are producing, then I guarantee with 100% certainty that you have not solved the Collatz conjecture or any other open problems you think you have solved. Please behave and act professionally.

Comment number 3, Taha Muhammad: Jul 15, 2026, 21:27

I will tell you "Thank you for your comment".

Response,
Peter M :
Jul 16, 2026, 02:23

You are not addressing any feedback or the basic mistakes I have pointed out. Your paper does not even contain a literature review, which makes it obvious you have not done any research on the twin prime conjecture.

Comment number 2, Taha Muhammad: Jul 15, 2026, 02:23

Thank you for your time, but you haven't mentioned a single step or idea. Are your words actually related to mathematics, or are they merely advice, presented in a way that is not even academic (Cambridge level, or what?).

Response,
Peter M :
Jul 15, 2026, 06:48

Here are some of the basic errors in your claimed proof: 1. You mention that x=11 does not produce a twin prime pair (3+2x,5+2x) = (25,27), but this does not mean that x is not a multiple of 11. When x=88, the pair (3+2x,5+2x) = (179,181) is a twin prime pair. 2. The fact that A is a subset of an infinite set does not imply that A is an infinite set, because finite sets are also subsets of an infinite set. 3. The statement A <--> TP ("A if and only if TP") is meaningless because A and TP are sets and not true/false statements. Do you mean to say A is the set of twin primes? Or the set of values x such that (3+2x,5+2x) is in TP? Either way, you have no argument that TP is infinite. Moreover, consider the fact that you are attempting a deceptively simple problem that has been unsolved for centuries, so elementary methods like yours almost certainly won't resolve it.

Comment number 1, Peter M: Jul 11, 2026, 15:46

This paper is riddled with extremely basic math and logical errors. Based on your repeated preprints claiming to solve famous open problems, this is just another crankish attempt at a proof and will never be accepted by any reputable journal or conference. I suggest actually doing your research because you are only wasting your time otherwise. This website is for scholarly content and yours is nowhere near that level.

Version History

Jul 11, 2026 Version 1

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DOI

10.33774/coe-2026-vp5c4
D O I: 10.33774/coe-2026-vp5c4 [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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