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Old 2020-10-29, 07:54   #34
kwalisch
 
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Default We have successfully verified PrimePi(1e28)

We have successfully verified PrimePi(1e28). I mistakenly posted the announcement in other thread: https://mersenneforum.org/showthread...248#post561248

Since we now have computed already many different prime counting function records (e.g. PrimePi(10^n), PrimePi(2^n), PrimePi(e^n), ...) and it is difficult to keep track of all these records in this forum I have created a Records.md page in primecount's repository on GitHub that contains all records computed so far. With the recent primecount improvements it is likely that more records will be added in the coming months.
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Old 2020-10-29, 10:04   #35
SethTro
 
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Quote:
Originally Posted by kwalisch View Post
We have successfully verified PrimePi(1e28). I mistakenly posted the announcement in other thread: https://mersenneforum.org/showthread...248#post561248

Since we now have computed already many different prime counting function records (e.g. PrimePi(10^n), PrimePi(2^n), PrimePi(e^n), ...) and it is difficult to keep track of all these records in this forum I have created a Records.md page in primecount's repository on GitHub that contains all records computed so far. With the recent primecount improvements it is likely that more records will be added in the coming months.
Fun page! Good luck extending these sequences!
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Old 2022-01-31, 11:51   #36
hal1se
 
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x=10^N π(x)
_____ ___
please note; comma is ,
for example, natural base is e=~ 2,7182818 ......

10^3 169
--} exp(1e3/169)=~371,35547715518192276458323785234 nbp
--} 1e3=~exp(6,9077552789821370520539743640531)
--} ln(nbp)=~ 5,9171597633136094674556213017751
difference; 0,990595515668527584598353062278

10^27 16352460426841680446427400
--} exp(1e27/16352460426841680446427400)=~ 361707074950517568846264313,78375 nbp
--} 1e27=~exp(62,169797510839233468485769276478)
--} ln(nbp)= 61,152876930896222750747748328791
difference; 1,016920579943010717738020947687

10^28 157589269275973410412739599?
--} exp(1e28/157589269275973410412739599)=~ 3619369212249243946780062787,5502 nbp?
--} 1e28=~exp(64,472382603833279152503760731162)
--} ln(nbp?)=~ 63,456097270733608293551439571667
difference; (64,472382603833279152503760731162)-63,456097270733608293551439571667=~
difference; 1,016285333099670858952321159495?

simple question: if N goes inf. then exp(10^N / π(10^N)) = nbp then the difference; ln(10^N) - ln(nbp) = ?

another sense, if someone imagine xor see:; the value nbp ;:
1e28's the so-called nbp value is, how?
if we sure nbp then any smart child, only handy calculation and a few seconds, windows' 7 decimal 32 digit calculator:
1e28/ln(nbp?)= 157589269275973410412739599
another simple question: do i know that the nbp value is the correct npb value? what do i know that i know?
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Old 2022-01-31, 13:53   #37
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Quote:
Originally Posted by hal1se View Post
x=10^N π(x)
_____ ___
please note; comma is ,
for example, natural base is e=~ 2,7182818 ......

10^3 169
--} exp(1e3/169)=~371,35547715518192276458323785234 nbp
--} 1e3=~exp(6,9077552789821370520539743640531)
--} ln(nbp)=~ 5,9171597633136094674556213017751
difference; 0,990595515668527584598353062278
<snip>
You are taking the logarithm of the exponential function. This is a roundabout way of doing nothing.

In general your "difference" is ln(x) - x/pi(x). So?

The most significant thing I see in your post is that your prime counts are too large by 1.
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Old 2022-03-04, 12:30   #38
kwalisch
 
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Default New prime counting function record: PrimePi(10^29)

David and myself are pleased to announce that we have computed a new prime counting function world record: PrimePi(10^29) = 1,520,698,109,714,272,166,094,258,063.

The computation took 92.76 CPU core years (only physical CPU cores are counted) and the peek RAM usage was 1.3 terabytes. We used the primecount-6.4 program with backup functionality for this computation. Half of the computation was run on my dual socket AMD EPYC server with two AMD EPYC 7642 CPUs with a total of 96 CPU cores (192 threads), here is a picture of it. The other half of the computation was run on David's dual socket AMD EPYC server with two AMD EPYC 7742 CPUs with a total of 128 CPU cores (256 threads). Overall our servers ran for about 6 months to complete that computation.

While computing prime counting function records is fun, having a major war just 1000 miles from where I live is no fun at all. David and myself like to express that we stand in solidarity with the Ukrainians and their democratically elected president Volodymyr Zelenskyy. We strongly condemn Russia's war of aggression against Ukraine without much regard for civilians. In 1940 Hitler Germany invaded and occupied my home country Luxembourg, our democratically elected government was replaced by a pro-German puppet regime and we were disallowed to speak our native Luxembourgish language. What Russia is currently doing in Ukraine is exactly the same. The United States, Europe and the other democracies from around the World must provide as much support as possible to Ukraine.

[Moderator's note August 2022: Moderators are not here to censor opinions. However, in the future please post political observations in the Soap Box subforum. Since this post has already been here for 5 months, I've decided to leave it here rather than move it to the Soap Box]

Last fiddled with by Prime95 on 2022-08-09 at 18:36 Reason: Note from moderators added
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Old 2022-03-04, 21:36   #39
mart_r
 
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Quote:
Originally Posted by kwalisch View Post
David and myself are pleased to announce that we have computed a new prime counting function world record: PrimePi(10^29) = 1,520,698,109,714,272,166,094,258,063.

Amazing!

\(\pi(10^{29}) = 3 \cdot 223 \cdot 820976459 \cdot 2768765559530353\)

The value \(Li(x)-\pi(x)\) is again very close to \(\sqrt{x}/\log(x)\) (for \(x=10^{29}\) of course). Riemann's approximation is about 14 times closer to \(\pi(x)\) than \(Li(x)\) (that is, \(\lvert\frac{Li(x)-\pi(x)}{R(x)-\pi(x)}\rvert \approx 13.55\)). May I again tentatively point to my question over at https://www.mersenneforum.org/showthread.php?t=26940?

Last fiddled with by mart_r on 2022-03-04 at 21:38
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Old 2022-03-05, 07:39   #40
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Kim and I have verified our result that PrimePi(10^29) = 1,520,698,109,714,272,166,094,258,063 using different starting parameters. The verification run was being done while the discovery run was completing.
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Old 2022-03-12, 18:57   #41
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Wow, big congratulations from me !
My page was updated: https://www.pzktupel.de/counting/PI_01.html
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Old 2022-08-09, 17:44   #42
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Quote:
Originally Posted by kwalisch View Post
David and myself are pleased to announce that we have computed a new prime counting function world record: PrimePi(10^29) = 1,520,698,109,714,272,166,094,258,063.
Congratulations!
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Old 2022-08-12, 10:48   #43
ThomasK
 
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Quote:
Originally Posted by kwalisch View Post
David and myself are pleased to announce that we have computed a new prime counting function world record: PrimePi(10^29) = 1,520,698,109,714,272,166,094,258,063.

The computation took 92.76 CPU core years (only physical CPU cores are counted) and the peek RAM usage was 1.3 terabytes. We used the primecount-6.4 program with backup functionality for this computation. Half of the computation was run on my dual socket AMD EPYC server with two AMD EPYC 7642 CPUs with a total of 96 CPU cores (192 threads), here is a picture of it. The other half of the computation was run on David's dual socket AMD EPYC server with two AMD EPYC 7742 CPUs with a total of 128 CPU cores (256 threads). Overall our servers ran for about 6 months to complete that computation.



Congratulations on finding pi(10^29). I guess the computational effort for this is of the order of O(n^0.6).

We are working on creating an total analysis for the minimum game up to 10^18. The computational effort for the total analysis minimum for all numbers up to n is between O(n * ln ln n) and O(n * ln n) With the computational effort you did for pi(10^29) you would almost reach 10 ^18 with the minimum total analysis. At 10^18 we assume the smallest Fünfwertzahl (five-value number).

The smallest Nullwertzahl ist 2, the smallest Einswertzahl is 3, the smallest Zweiwertzahl is 19, the smallest Dreiwertzahl is 173 and the smallest Vierwertzahl ist 3976733.


At the moment the minimum total analysis up to 10^12 is completely available.

If you are interested in dealing with the minimum problem, then you are welcome to take a look in our thread of the minimum problem:

https://www.mersenneforum.org/showthread.php?t=27056


Greetings from München,
Thomas
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