Did I solve a 150+ year old mathematical problem today?
The highlight of my day(/week/year/Century/life), as a couple of fellow regular contributors at the Unsolved Problems public Yahoo Group believed that a paper I submitted had solved a 150+ year old unsolved mathematics problem: https://groups.io/g/UnsolvedProblems/message/6132
https://groups.io/g/UnsolvedProblems/message/6145
However, based on subsequent messages in the group today by others, it looks like my paper may not actually have solved it! The discussion continues...
The text of the discussion:
6166 RE: [UnsolvedProblems] Goldbach and Twin Prime Conjectures: Proposed Proof Papers
Flatow, Daniel (NIH/NCI) [E]
Today at 7:49 PM
I agree! I read the proof, and it looks fine to me. I can’t believe that I am saying this!
From: UnsolvedProblems@yahoogroups.com [mailto:UnsolvedProblems@yahoogroups.com]
Sent: January 25, 2017 2:28 AM
To: UnsolvedProblems@yahoogroups.com
Subject: RE: [UnsolvedProblems] Goldbach and Twin Prime Conjectures: Proposed Proof Papers
Hi Vaskor,
So sorry for my delay in posting, I just got to read your updated proof a few hours ago.
Nicely done finding the error and reducing the twin primes by a factor of 6!
I see that Lee has chimed in and presented a shorter and quite elegant version of your basic method of proof.
I can't see any error in your proof, and neither apparently can Lee, who is the most meticulous mathematical scribe I have come across!
So - and I can't believe I am witnessing this - you Vaskor have proven of the Twin Prime Conjecture to the best of my knowledge !
Your numerator shifting seems to be the simple key, the trick, that has somehow eluded the best mathematicians fo r over 150 years.
Truly, this seems like a dream. Vaskor, I don't know how you are sleeping at night!!
Please upload your proof to Tim's Unsolved Problems site, a proof refined perhaps with the collaboration of Lee who has a lot of experience in mathematical presentation. If you are a student and short on funds, I would be happy to pay the fee to put it on Tim's site.
Congratulations!
Mark
PS Understandably, people will think this premature. & nbsp;And it is. But I just can't help myself !
---In UnsolvedProblems@yahoogroups.com, <vaskorb@...> wrote :
Hi Mark,
No problem at all, and thank you very much for your further comments, which are much welcomed!
Your analysis below led me to discover a mistake I had made in paragraph 10, which eventually led to the erroneous result you identified. Fortunately, it looks to me that it is possible to remedy the mistake, which I have now done. The corrected paper is attached.
To correct the problem, I had to use a technique relating to inequalities we learnt in the second year of our Mathematics course, which made a proof a bit longer. As a result, I have split paragraph 10 into several separate paragraphs for easier referencing. I have not made any change s to other paragraphs, so you need not read these again, unless you wish to scrutinise them again in more detail.
Vaskor Basak
---In UnsolvedProblems@yahoogroups.com, <mark.underwood@...> wrote :
Hi Vaskor
First, my apologies because I posted in haste last night because I had to hurry off.
Your q was very well defined and I understood it as you presented it.
I really liked how you did the numerator shift one to the left.
Now, when I am examining your proof less hurriedly, my conclusion about your q being two things was mistaken, and you are correct.
Frankly, last night when I saw this line near at the e nd of your proof:
1. Therefore, there must be more than 3(6n+1)^(1/2) twin primes less than or equal to 6n+1.
I 'knew' it couldn't be correct. And indeed, it is not correct when n is lower. But when I checked this morning, it actually becomes correct for n > 7139 (!)
So now I'm doubly confused! I can't find a flaw in Vaskor's proof, and thus I can't see why the result is incorrect for lower n ! But I'm distracted and doing other things now. I welcome other people to scrutinize Vaskor's proof. I hope to look at it again later today...
Mark
---In UnsolvedProblems@yahoogroups.com, <vaskorb@...> wrote :
Hi Mark,
Thank you very much for looking through the paper and for your comments.
Thank you also for your well considered reply in relation to paragraph 10.
In the first sentence of the extract from the paper included in your message below, you said that "q is the prime factors screened out". Did you mean "q is the number of prime factors screened out"? If so, that is not actually correct. What I meant in the paper was that 2/q is the maximum proportion, as a fraction, of potential twin prime rows screened out in the final stage only, for the prime number q. This is similar to, for example, 2/17th of the potential twin prime rows being screened out when considering the prime number 17, at an earlier stage.
In the second sentence, q is not in fact 6n+1. q is the highest prime less than or equal to (6n+1)^(1/2). Thus, the expression in the second sentence might look something like 3/5.5/7.9/11.11/13...51305/51307, for example, if 51307 happens to be the highest prime less than or equal to (6n+1)^(1/2).
Vaskor Basak
---In UnsolvedProblems@yahoogroups.com, <mark.underwood@...> wrote :
Hi Vaskor
Your reasoning is very clear and well done. The problem is in part 10, regarding what q means.
Here is where q gets a double meaning:
This process is continued until finally a maximum of 2/q are struck out when striking out multiples of q. Therefore the ratio of rows remaining at the end of the whole process must be at least 3/5.5/7.9/11.11/13…(q-2)/q
In the first sentence above, q is the prime factors screened out, and is less than sqrt(6n+1). But in the second sentence, q is actually 6n+1. This mix up messes with your conclusion as you will easily see!
Mark
---In UnsolvedProblems@yahoogroups.com, <vaskorb@...> wrote :
Whilst on the subject of the Goldbach and Twin Prime conjectures, it occurred to me that I had written a paper on a proposed approach for each in 2011, neither of which I had ever got around to sharing with this group.
I am therefore uploading the Twin Prime Conjecture paper now, with the hope of stimulating further discussion on approaching this problem.
I intend to do the same for the Goldbach Conjecture paper later.
Vaskor Basak