20200511, 13:21  #1 
"murat"
May 2020
turkey
53 Posts 
need explanation about gap of primes
I need an explanation . what does the meaning of "the prime pairs with the distence od 70000000
it means any consecutive two piimes has the diatance of less than 70000000 or a pair of prime has the distance leas than 70000000 can be have big distance than 70000000 to consecutive prime 
20200511, 16:51  #2 
Einyen
Dec 2003
Denmark
3·23·43 Posts 
It means that even though the average gap between primes goes towards infinity for larger and larger numbers there will always be more pairs of primes less than 70,000,000 apart further out the number line.
The bound 70,000,000 by Zhang was lowered to 246 by the Polymath Project and James Maynard, and even to 12 or 6 if the Elliott–Halberstam conjecture is true or true in its generalized form: https://en.wikipedia.org/wiki/Prime_gap#Upper_bounds So there are infinitely many prime gaps lower than 246 at least. 
20200511, 17:55  #3 
"murat"
May 2020
turkey
53 Posts 
thank you so much . I need a calculated gap until a giant number
for example maximum gap is 1000000 until 10^32 or just like no big gap than 10^7 until 10^40 .. . do you know any study about it 
20200512, 00:22  #4 
Feb 2017
Nowhere
3571_{10} Posts 
Alas, the only way I know of to be certain of the largest gap between successive primes up to X is by finding all the primes up to X and checking. This has been done up to 4x10^{18} according to this page.
I wouldn't hold my breath waiting for an exhaustive check up to 10^{32}. I would consider a gap of size 10^{7} between consecutive primes < 10^{32} to be extremely unlikely (see below). The maximal gap between primes < X is conjectured to be something like ln^{2}(X). Based on the maximal gaps table and those conjectures, I would expect a maximal gap between consecutive primes < 10^{32} of around 4500. For a gap of size 10^{7}, the conjectures indicate the smaller prime would have to be of size around 10^{1373} or 10^{1376} or something like that. 
20200512, 01:08  #5 
Einyen
Dec 2003
Denmark
3·23·43 Posts 
I'm confused by the Ford–Green–Konyagin–Maynard–Tao formula:
https://en.wikipedia.org/wiki/Prime_gap#Lower_bounds Gn > c * (log n * log log n * log log log log n / log log log n) So infinitely many gaps are larger than this expression for some constant c>0. But how does this help when there is no restriction on how small the constant c is? Last fiddled with by ATH on 20200512 at 01:08 
20200512, 02:04  #6  
Feb 2017
Nowhere
DF3_{16} Posts 
Quote:
Also, a fairly recent preprint has an even better result. LARGE GAPS BETWEEN PRIMES by JAMES MAYNARD dated 28 October 2019 Quote:


20200512, 15:38  #7  
Einyen
Dec 2003
Denmark
2967_{10} Posts 
Quote:
The paper from 2019 seems to be the same result as the one from 2014 ? I do not claim to understand anything in these papers. Edit: It seems to be the same paper as 2014, they even show it at 6:08 in this Numberphile video: https://www.youtube.com/watch?v=BH1GMGDYndo Last fiddled with by ATH on 20200512 at 15:41 

20200513, 02:22  #8  
Feb 2017
Nowhere
DF3_{16} Posts 
I can't follow the arguments, but the result stated in the abstract is what it is. The logloglog term in the denominator is squared.
As it it in another paper I found, LARGE GAPS BETWEEN CONSECUTIVE PRIME NUMBERS by KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, AND TERENCE TAO, with a result similar to that in the preprint I linked to. Quote:


20200513, 05:51  #9 
Einyen
Dec 2003
Denmark
3×23×43 Posts 
Check page 15 and 18 in the pdf you linked:
https://terrytao.files.wordpress.com/2015/07/lat.pdf Page 15 shows the Rankin formula from 1938 which is the one with (log log log n)^{2} in the demoninator for which Ford–Green–Konyagin–Maynard–Tao proved c can be as large as you want earning the $10,000 prize from Erdos. Page 18 shows their improved formula from Aug 2014 which is with (log log log n) in the denominator. Their formula is in the Numberphile video at 8:20: https://www.youtube.com/watch?v=BH1GMGDYndo&t=8m20s Last fiddled with by ATH on 20200513 at 06:09 
20200513, 05:55  #10  
Jun 2003
Oxford, UK
1897_{10} Posts 
Quote:


20200513, 12:00  #11  
Feb 2017
Nowhere
3,571 Posts 
Quote:
Got it! LONG GAPS BETWEEN PRIMES by KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, JAMES MAYNARD, AND TERENCE TAO Quote:


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