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Old 2020-05-11, 13:21   #1
drmurat
 
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Default 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
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Old 2020-05-11, 16:51   #2
ATH
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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.
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Old 2020-05-11, 17:55   #3
drmurat
 
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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
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Old 2020-05-12, 00:22   #4
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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 4x1018 according to this page.

I wouldn't hold my breath waiting for an exhaustive check up to 1032.

I would consider a gap of size 107 between consecutive primes < 1032 to be extremely unlikely (see below).

The maximal gap between primes < X is conjectured to be something like ln2(X).

Based on the maximal gaps table and those conjectures, I would expect a maximal gap between consecutive primes < 1032 of around 4500.

For a gap of size 107, the conjectures indicate the smaller prime would have to be of size around 101373 or 101376 or something like that.
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Old 2020-05-12, 01:08   #5
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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 2020-05-12 at 01:08
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Old 2020-05-12, 02:04   #6
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Quote:
Originally Posted by ATH View Post
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?
The logloglog term in the denominator should be squared. Actual numerical values for c had been obtained. See this page.

Also, a fairly recent preprint has an even better result.

LARGE GAPS BETWEEN PRIMES by JAMES MAYNARD dated 28 October 2019

Quote:
Abstract. We show that there exist pairs of consecutive primes less than
x whose difference is larger than

t(1+o(1))(logx)(loglogx)(loglogloglogx)(logloglogx)−2

for any fixed t. This answers a well-known question of Erdős.
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Old 2020-05-12, 15:38   #7
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Quote:
Originally Posted by Dr Sardonicus View Post
The logloglog term in the denominator should be squared. Actual numerical values for c had been obtained. See this page.
The c-values in page 16 is for the 1938 Rankin formula with (log log log n)2 in the denominator. The improvement by Ford–Green–Konyagin–Maynard–Tao (2014) in page 18 has only (log log log n) in the denominator.

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 2020-05-12 at 15:41
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Old 2020-05-13, 02:22   #8
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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:
ABSTRACT. Let G(X) denote the size of the largest gap between consecutive primes below X . Answering a question of Erdős, we show that

G(X)\;\ge\;  f(X)\frac{log X loglog X loglogloglog X}{(logloglog X)^{2}}\;\text{,}

where f(X) is a function tending to infinity with X. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.
So I'm pretty sure that logloglog term is supposed to be squared. If a result were known where it wasn't, and the other factors were the same, it would be a stronger result, and nobody would continue with the (logloglogX)2 results.
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Old 2020-05-13, 05:51   #9
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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 2020-05-13 at 06:09
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Old 2020-05-13, 05:55   #10
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Quote:
Originally Posted by Dr Sardonicus View Post
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 4x1018 according to this page.
This very group ("Mersenneforum Prime Gap Searches") continued the search up to 2^64.
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Old 2020-05-13, 12:00   #11
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Quote:
Originally Posted by ATH View Post
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
Ahh... Thanks, missed that... I should be able to find the actual paper. <google google, toil and trouble...>

Got it! LONG GAPS BETWEEN PRIMES by KEVIN FORD, BEN GREEN, SERGEI KONYAGIN, JAMES MAYNARD, AND TERENCE TAO

Quote:
ABSTRACT. Let pn denote the n-th prime. We prove that

\max_{p_{n+1}\le X}  (p_{n+1} - p_{n}) \gg \frac{logXloglogXloglogloglogX}{logloglogX}

for sufficiently large X, improving upon recent bounds of the first three and fifth authors and of the fourth author. Our main new ingredient is a generalization of a hypergraph covering theorem of Pippenger and Spencer, proven using the Rödl nibble method.
See also this discussion in Terence Tao's updates.
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