I need a cost estimate for some data.

[NBtarheel_33]

I get the feeling that I have wasted far more time on this thread than any wise person ought to, but in the spirit of helping Ms. Sophrosyne to be heard:

Term    Pn.    Mp.                       Greater
0.          0.      1.                           3.5311288...
1           1.      2.                           8
2           2.      3.                         36
3.          3.      4.                       666
4.          4.      6.                 222111
5.          6       8.      24666759216
6.          8.      9.                            3.04E             20
7.          9.    12.                            4.62E             40
8.        12.    12.                            1.07E             81
9.        12.    13.                            5.73E           161
10.      13.    14.                            1.64E           323
11.      14.    15.                            1.34E           646
12.      15.    17.                            9.11E         1291
13.      18.    20.                            4.14E         2583
14.      20.    23.                            8.61E         5166
15.      23.    26.                            3.70E       10333
16.      26.    27.                            6.87E       20666
17.      27.    30.                            2.36E       41333
18.      30.    31.                            2.78E       82666
19.      31.    31.                            3.89E     165332
20.      31.    32.                            7.57E     330664
21.      33.    33.                            2.86E     661329
22.      33.    37.                            4.11E   1322658
23.      34.    38.                            8.44E   2645316
24.      38.    39.                            3.56E   5290633
25.      39.    44.                            6.36E  10581266
26.      44.    47 ?                          2.02E  21162533
27.               47?                           2.05E  42325066
28.                                                2.10E  84650132
29. (?) 2'346348367-1                2.22E 169300264
30. if Mp then Pn                        2.46E 338600528
31.                                                3.40E 677201056

[NBtarheel_33]

Well, it ain't pretty, but it's readable...

Now, Kathegetes, please return my good deed by explaining what's going on in that table.

[CRGreathouse]

Originally Posted by Kathegetes

Is it correct that 6592875536327244522 is limit to sieved in completed order?

I can't find the exact value for π(304224505122393846936) if that's what you mean. My current resources limit me to 2⁶³, and that's about 2⁶⁸. (Does anyone here have better tech for this?)

[danaj]

I get the same Dusart 2010 estimates you got, albeit we round differently (I decided to be more conservative especially concerned about FP accuracy in general).

There are three good open source prime count software I know of, mine in MPU is limited to 2^64, Christian Bau's old code should work to 2^64, and Kim's newest D-R routines in primecount are limited to 2^63. Mine takes about 8.3 minutes on a single core for 10^18, Kim's takes about 1.5 minutes with 8 threads. We've both given some thought to extending them, probably using gcc __int128 types, but no code yet. Obviously Oliveira e Silva and others have code that goes higher but they don't release their code.

[Kathegetes]

Originally Posted by NBtarheel_33

Please note, as was discussed here over two (Earth) years ago, the Mersenne number is not, cannot, and never will be prime. So if you are basing any theories or conjectures on the primality of that number, as your table seems to indicate, they will be mistaken.

Was my hero because of kindness. Poor teacher happy to save electric. Wrote story of same.
I see the table is readable. Sorry it got shrimped to the left.
Is newest perfect less than term 27 = 2.053355...10'42325066 ?
I do not need to know but if you please.

[NBtarheel_33]

Originally Posted by Kathegetes

Was my hero because of kindness. Poor teacher happy to save electric. Wrote story of same.
I see the table is readable. Sorry it got shrimped to the left.
Is newest perfect less than term 27 = 2.053355...10'42325066 ?
I do not need to know but if you please.

The largest known (and newest, being associated with the discovery of M48 in early 2013) perfect number is . This has 34,850,340 digits, which is on the order of , which is much, much, much less than .

[Kathegetes]

Originally Posted by NBtarheel_33

Well, it ain't pretty, but it's readable...

Now, Kathegetes, please return my good deed by explaining what's going on in that table.

It is the count of Pn that may be the wonder for children for the future. Maybe there are 5 or 6 in the last term. I like the pattern in powers of ten by 8 tri.... too. Also I see the greatest prime in the work given (please complete to check if holds) produced its P n in the next term.
Any Neils Bohr said, "Prediction is difficult, especial the future."
Up up up down. Up up up up down. Up up up up up ? But if it doesn't go back down it must keep going up. So we do the fun. I predict at some term 28 Pn. Maybe 6 already.
The count ÷ term? More numbers more patterns. Instead of 8 just the primes can be chased.
Time? Thanks all of you. We are at a moment in the history of Pn which you can do fun math about. I use letters to reduce all the p Mp Pn ...peee.
In what year is the fun quaesita ? At present half the K primes are members of set t and half are set d .
Set t goes ( 2 and 6n-1...
Set d goes (3 and 6n +1...
When will it be so in earth years again?
I teach it this way.
Let all p form 6n-1 be a t
All p form 6n+1 be a d
All t and all d and 2 and 3 be some k.
All 2'k-1 be a M
When M is prime it be a D upper case member of d
Then k is a K (Known)
D (D +1) ÷2 be a S member of lower case s, and sum of parts.
I have a pretty little proof in a flower and some triangles.
As for content ? Please don't throw me in the briar patch.
I

[Uncwilly]

Originally Posted by Kathegetes

Up up up down. Up up up up down. Up up up up up ?

Do you mean:
↑↑↓↓←→←→BA?

[CRGreathouse]

Originally Posted by danaj

I get the same Dusart 2010 estimates you got, albeit we round differently (I decided to be more conservative especially concerned about FP accuracy in general).

I wrote a general routine which wraps the Dusart estimates along with a large number of others and handles appropriate (I hope!) rounding. that way I can think about it fairly carefully upfront and not worry too much each time I use it.

Originally Posted by danaj

There are three good open source prime count software I know of, mine in MPU is limited to 2^64, Christian Bau's old code should work to 2^64, and Kim's newest D-R routines in primecount are limited to 2^63.

Very nice. I've been using Christian Bau's code -- and unless I've done something wrong setting it up it looks like it works up to 2^63. I think I'll give yours a try.

Originally Posted by danaj

Mine takes about 8.3 minutes on a single core for 10^18, Kim's takes about 1.5 minutes with 8 threads.

Sounds like a big performance improvement over the Bau code -- it takes me over 30 minutes on a single core (admittedly, my machine is probably slower).

Originally Posted by danaj

We've both given some thought to extending them, probably using gcc __int128 types, but no code yet.

Cool, I'd like to see that when/if it happens.

[Kathegetes]

Originally Posted by CRGreathouse

I can't find the exact value for π(304224505122393846936) if that's what you mean. My current resources limit me to 2⁶³, and that's about 2⁶⁸. (Does anyone here have better tech for this?)

America has a atom clock for time synchronous many lands and GPS dateliyes in space. Does she have same asfor primes in order and time? Try again. Who do you look to for primes in count and order?

[danaj]

Originally Posted by CRGreathouse

I wrote a general routine which wraps the Dusart estimates along with a large number of others and handles appropriate (I hope!) rounding. that way I can think about it fairly carefully upfront and not worry too much each time I use it.

I do as well, but I was nervous about C libraries (logl) and floating point accuracy on random machines. Admittedly I'm probably not nearly paranoid enough for that, so I should just do the calculations assuming reasonable double-precision. It just makes my numbers one too far, which I viewed as a reasonable sacrifice at the time. Most of my effort was spent on tightening the bounds for smaller inputs.

Very nice. I've been using Christian Bau's code -- and unless I've done something wrong setting it up it looks like it works up to 2^63. I think I'll give yours a try.

I haven't run it in quite a while but I thought he had values up to 1.8e19 in his test program. I used a lot of the test values from SAGE ticket 7539 to test some values between 2^63 and 2^64.

Sounds like a big performance improvement over the Bau code -- it takes me over 30 minutes on a single core (admittedly, my machine is probably slower).

Some of it is my using a fast machine (4.3GHz i4770K), some is how sensitive his code is to all the myriad tuning factors. I use the basic algorithm from his paper, so it's all in the tuning factors and the small routines. In particular the prefix sum counting is quite different. Also I highly recommend compiling with -march=native -- that should enable the asm popcnt if available. Without that, I use my own C popcount, which is much faster than gcc's non-asm code but not as good as the asm. Sadly knowing whether the asm is available or not is a very non-portable runtime check.