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2017-12-28, 17:45   #67
robert44444uk

Jun 2003
Oxford, UK

190810 Posts

Quote:
 Originally Posted by danaj New record merit gap from Gapcoin: 8350 41.93878373 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227 The first one with merit 40+.
Oh my - this is a very significant result and will be very hard to beat - 40 was a major target and this result blows it away.

Maybe this will be the next challenge for the PGS when we get to 2^64.

 2018-01-01, 00:52 #68 George M   Dec 2017 2·52 Posts I love how we all think the prime numbers are random when really.... I don’t think a lot of people know about the Prime Gap Equation but I found it on a Wikipedia Article and it just shows that every prime number determines the following prime number. So, WHAT ARE WE ALL TALKING ABOUT?! heh. And I read a book by Australian mathematician and stand-up comedian Matt Parker, called “Things to Make and Do in the Fourth Dimension” and he says that for some prime number p, there exists another prime number q that ranges from (p, p + 5414). Well, this is how I’m phrasing it, but he simply said in the book that each gap between two adjacent (neighbouring) prime numbers have an upper bound of 5414. Last fiddled with by George M on 2018-01-01 at 00:56 Reason: Because people need to know about the bound of prime gaps.
2018-01-01, 02:08   #69
VBCurtis

"Curtis"
Feb 2005
Riverside, CA

450010 Posts

Quote:
 Originally Posted by George M Well, this is how I’m phrasing it, but he simply said in the book that each gap between two adjacent (neighbouring) prime numbers have an upper bound of 5414.
We don't pay much attention to false claims, so you're right that we don't know about this equation nor his false claim.

For your own education, find the next prime after this number:
293703234068022590158723766104419463425709075574811762098588798217895728858676728143227

Hint: the next prime is more than 5414 greater than this number, which disproves the hooey you cite.

2018-01-01, 03:52   #70
axn

Jun 2003

4,789 Posts

Quote:
 Originally Posted by VBCurtis We don't pay much attention to false claims, so you're right that we don't know about this equation nor his false claim.
I think OP misunderstood what is being claimed.

Basically, there was a result that there are infinitely many prime pairs p,q such that the gap q-p is bounded by a small number. They successively improved the upper bound on that gap (which at one point stood at 5414 -- See http://michaelnielsen.org/polymath1/...ime_gap_bounds)

 2018-01-01, 11:39 #71 danaj   "Dana Jacobsen" Feb 2011 Bangkok, TH 2·3·151 Posts I don't think we need to add any more on the recent topic. Based on George's other posts today, I think it was an early New Year's celebration that included random posts to lots of threads. On topic, 10 of the top 14 merits were found in 2017 including the top 5. Before I moved resources over to the PGS exhaustive search, I'd put a fair amount into smaller P1s, leading to a lot more large-merit finds. Gapcoin may do even more in 2018 given the popularity of cryptocoins these day
2018-01-01, 13:05   #72
George M

Dec 2017

2·52 Posts

Quote:
 Originally Posted by VBCurtis We don't pay much attention to false claims, so you're right that we don't know about this equation nor his false claim. For your own education, find the next prime after this number: 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227 Hint: the next prime is more than 5414 greater than this number, which disproves the hooey you cite.
But... but... GOD DAMMIT. Let’s just keep the upper bound of the gap at 70,000,000 where it originally was at..

 2018-01-01, 14:02 #73 George M   Dec 2017 1100102 Posts Prime Gap Hystory On 13 May 2013, an upper bound of prime gaps was proven to be 63,374,611 (rounding to 70 million). This was done by Yitang (Tom) Zhang. Then Tim Trudgian brought it down to 59,874,594 with Scott Morrison bringing it further down to 59,470,640 around late May. At 31 May however, it was brought down to 42,342,946. Then a mathematician called Terence Tao who learnt algebra at aged 3, completed his maths degree at aged 16, got a maths PhD and won a Fields Medal in 2006, brought down the bound to 42,342,924. Terence Tao is known as the “hyper-genius” at maths with an IQ of 220 (world’s highest). He and another Fields Medalist, Tim Gower, then started an open project as part of Polymath where mathematicians could join together and collaborate to bring this bound down. As of 20 July 2013, the upper bound was brought down to 5414. Doesn’t sound hooey to me, but if you say so...
2018-01-01, 14:11   #74
10metreh

Nov 2008

2·33·43 Posts

Quote:
 Originally Posted by George M On 13 May 2013, an upper bound of prime gaps was proven to be 63,374,611 (rounding to 70 million). This was done by Yitang (Tom) Zhang. Then Tim Trudgian brought it down to 59,874,594 with Scott Morrison bringing it further down to 59,470,640 around late May. At 31 May however, it was brought down to 42,342,946. Then a mathematician called Terence Tao who learnt algebra at aged 3, completed his maths degree at aged 16, got a maths PhD and won a Fields Medal in 2006, brought down the bound to 42,342,924. Terence Tao is known as the “hyper-genius” at maths with an IQ of 220 (world’s highest). He and another Fields Medalist, Tim Gower, then started an open project as part of Polymath where mathematicians could join together and collaborate to bring this bound down. As of 20 July 2013, the upper bound was brought down to 5414. Doesn’t sound hooey to me, but if you say so...
These are not upper bounds on gaps. Zhang proved that there are infinitely many prime gaps smaller than 70,000,000. This does NOT mean that all gaps are smaller than 70,000,000.

In fact arbitrarily large gaps exist: n!+m is divisible by m for m ≤ n, so there are n-1 consecutive composite numbers from n!+2 to n!+n. This gives a gap of size at least n.

2018-01-02, 07:19   #75
George M

Dec 2017

2·52 Posts

Quote:
 Originally Posted by 10metreh These are not upper bounds on gaps. Zhang proved that there are infinitely many prime gaps smaller than 70,000,000. This does NOT mean that all gaps are smaller than 70,000,000. In fact arbitrarily large gaps exist: n!+m is divisible by m for m ≤ n, so there are n-1 consecutive composite numbers from n!+2 to n!+n. This gives a gap of size at least n.
Oh. And I misspelt “history” btw... but anyway, thanks for that clarification.

 2018-01-03, 19:35 #76 mart_r     Dec 2008 you know...around... 25·19 Posts The new Gapcoin discovery is a marvel. It's reminiscent of the Nyman gap of 1132. To top it off: if the left-hand bounding prime was composite, it would expand to a gap of merit=46.71 - which would have been an even more mind-blowing result. And all that without the benefits of a large primorial. I've attached a graph that shows that the numbers in the primeless interval that are coprime to about 200# (the order of magnitude of the primes themselves) is even a bit above the average. The graph itself shows that Gapcoin indeed uses "random" numbers, that is to say, without using primorials to take advantage of cancelling out a lot of small factors. (I've just come up with the term "coprime profile" for it - catchy/appropriate?) - For comparison, the second graph shows the same for a gap that utilizes a primorial. Attached Thumbnails     Last fiddled with by mart_r on 2018-01-03 at 19:41
2018-01-03, 19:57   #77
CRGreathouse

Aug 2006

2·2,969 Posts

Quote:
 Originally Posted by 10metreh These are not upper bounds on gaps. Zhang proved that there are infinitely many prime gaps smaller than 70,000,000. This does NOT mean that all gaps are smaller than 70,000,000. In fact arbitrarily large gaps exist: n!+m is divisible by m for m ≤ n, so there are n-1 consecutive composite numbers from n!+2 to n!+n. This gives a gap of size at least n.
Right. To drive the point home: asymptotically more than 99% of primes are followed by gaps of length more than 70 million.

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