20180319, 06:21  #265  
Feb 2017
245_{8} Posts 
Quote:
Myself, I am only going to attempt a "proof" for the twin prime conjecture which I think works, not withstanding that it deals in the realms of infinity. Are you perhaps suggesting that I should choose the 1st of April, instead of the 17 of April! :) I only chose 17 of April for sentimental reasons because it is my daughter's birthday on that day. 

20180319, 06:30  #266  
Feb 2017
10100101_{2} Posts 
Quote:
About Goldbach conjecture and other things math I am really very clueless...my big problem is that I am a "prime" hobbyist, and that I have always considered prime numbers to be "ordered" rather than "random". A pattern for Primes of course does exist...a forward pattern! as delivered by operation of the Sieve of Eratosthenes. It is only a quest to find the reverse pattern! Last fiddled with by gophne on 20180319 at 06:34 Reason: Left out part of a sentance. 

20180319, 08:15  #267  
Aug 2006
5,987 Posts 
Quote:
The Prime Number Theorem showed the ("trivial") estimate that the gap following a prime p is infinitely often at most length log p or so. Sieve theory was developed to work on problems like the twin prime conjecture, and the Brun sieve (1915) was developed to prove that the sum of the reciprocals of the twin primes converged. Westzynthius made the first improvement over the trivial bound, with further (slow) progress through the 1930s by Erdős and Rankin. Linnik developed sieve theory further with the large sieve in 1941. It uses Fourier analysis and analytic number theory to allow larger sieving sets. The Selberg sieve was also developed around this time, this one being a combinatorial sieve rather than analytic. Careful choice of sieve weights are essential in this method. Using the large sieve, Barban, Bomberi, and Vinogradov proved a theorem which gives a Riemann Hypothesistype error bound to primes in arithmetic progressions when the modulus is up to roughly square root size. The ElliottHalberstam conjecture (EH) says that the modulus can be taken almost up to the size of the bounding variable itself. This kind of information would be useful in a proof of the twin prime conjecture. Chen's theorem (1966) was the first nearproof of the twin prime conjecture, limited by the parity problem. It proves that there are infinitely many primes p such that p+2 is either prime or semiprime. Ross published a simplification using the Selberg sieve. Fouvry & Iwaniec (1980) published an improved version of Bomberi and Vinogradov giving some information above the square root barrier, which in principle makes the twin prime problem attackable. The FriedlanderIwaniec theorem (1997) was essentially the first work to begin to break down the parity barrier, using a great deal of analytic number theory along with combinatorics and Fourier analysis. Various subsets of Goldston, Graham, Motohashi, Pintz, and Yıldırım (~2005; two examples) prove many results about small gaps between primes. One example: On EH, there are infinitely many primes p followed by a gap of at most 16. Sieve weights and Selberg diagonalization are tools. Green & Tao (2006) published probably the last major twin primerelated paper before the Zhang bomb dropped, and it gives an idea of what the state of knowledge was at that time. Zhang's theorem, at its heart, is an improvement similar to Fouvry & Iwaniec along with the ingredients needed to apply it to the bounded gap problem. It proves that there are infinitely many primes p followed by a gap of at most 70000000. It may be the largest single step we've taken toward the twin prime conjecture. The polymath project simplified and improved Zhang's proof and reduced the gap to ~2000. Maynard added some new ingredients to the proof which allowed a further reduction, then followed up with a new paper with the Tao et al. Polymath project to give a bound of 246 (and a bound of 6 on a stronger form of EH). Ford, Konyagin, Maynard, Pomerance, & Tao (2018) published a paper, improving on other recent ones I'm too tired to survey at the moment, on long gaps between primes. There seems to be a duality between short gaps and long gaps and this team seems to be unraveling the mystery. So there really is a lot out there, if you have the patience to learn it. I have quite a lot of learning to do myself. Edit: Here's a very rough chart of the connections between the steps leading us to current progress toward the twin prime conjecture. All mistakes are mine. Last fiddled with by CRGreathouse on 20180319 at 20:50 

20180410, 11:59  #268 
Feb 2017
3·5·11 Posts 
Twin Prime Conjecture
Hi CRGreathouse
Thank you for all the information in your post about work done in the area of a possible proof of the Twin Prime Conjecture and related gaps between prime numbers. I will try to look up some of these works to see if I am not going down a path which has already been researched (professionally) and has not managed to nail down a proof for the TPC, as not to be a bore. However, I still intend to post an attempted "proof" on the 17th as I have promised. My attempt would be based on the Sieve of Eratosthenes, Euclid's proof of the infinity of prime numbers, and a logical deduction based on the relationships/statements made/derived from the system of the "proof". What I would like to know is if it is possible to cut & paste text and tables from Excel & Word on this platform, as I have worked mostly in Excel. I hope that if my "proof" is not valid, that at least that it would have been a novel approach to the problem. Even at this late stage I am still struggling with my "logical deduction" on which the proof is based, since to me it appears sound, but there is a fogginess to it as one considers the abstraction of the logic at numbers that cannot be checked in normal computer time (using Excel or SAGE!). The "proof" is not very long/extensive! SO would not waste too much of people's brain power! 
20180410, 16:35  #269  
Aug 2006
1011101100011_{2} Posts 
Quote:
Here's my rough scale for grading purported proofs of the twin prime conjecture: 1,000,000 points for a correct proof of the twin prime conjecture 100 points for a clear attempted proof with interesting, nontrivial, correct, and possible publishable theorems/lemmas 0 points for a clear attempted proof 10 points for a vague, hardtofollow, or similar attempted proof 11 points for an attempted proof that is unsalvageable or 'not even wrong' 11 points for the variants of the standard proof The average score here, unfortunately, is less than 10. 

20180410, 22:17  #270  
Feb 2017
165_{10} Posts 
Quote:


20180410, 22:31  #271  
"Forget I exist"
Jul 2009
Dartmouth NS
2×3×23×61 Posts 
Quote:


20180424, 03:18  #272 
Feb 2017
3·5·11 Posts 
Hurdle
Struggling with my "logical deduction" to show that the methodology I use to locate and predict the location of twin primes, would be applicable at values approaching infinity, or more importantly in terms of the requirements of a proof, that twin primes could still be occuring approaching infinity. If I cannot resolve this shortly, then I will post the "imperfect" proof in order for others to perhaps debunk the approach from their vantage point, if anybody might be interested to do so.
I shall post on Google Docs as suggested. Apologies for not posting on the date intended, but I shall post what I have done in a few days time, even if I cannot resolve the problem (proof) myself, just for comment as to the worth or nonworth of the attempt/approach. 
20180424, 13:16  #273 
Aug 2006
5,987 Posts 

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