Hello!

>10^3999 + 4771 is prime.

Congratulations!!! :-)))

17 weeks is much smaller than the time calculated using
Giovanni's estimator.

Was the report file (*.cr) created?

Thanks,

Andrey

> So, Paul, in order not to do any work duplication, do you have
> a list of the proven primes you found?

Yes, they appear in the list I posted here a few months ago.

Rather than cluttering up the entire membership's mailboxes with a
duplicate posting, I'll send a personal copy.


Paul

An incredible record! Well done!

I had already noticed that TX2.1 was much faster, but
this is really fast.

Will you make a new estimator?

Bouk.

--- Giovanni La Barbera <giolaba@...> wrote:
> Hi,
>
> 10^3999 + 4771 is prime.
> The proof took about 3000 h of  a PENTIUM III, 800
> MHZ, with the help
> af a second PIII for difficult steps.
>
> The program used is Titanix  by Marcel Martin.
>
> Please see:
>
> http://www.znz.freesurf.fr/pages/titanixrecord.html
>
> Giovanni & Marco La Barbera
>
>
> Unsubscribe by an email to:
> primenumbers-unsubscribe@egroups.com
> The Prime Pages : http://www.primepages.org
>
>
>
> Your use of Yahoo! Groups is subject to
> http://docs.yahoo.com/info/terms/
>
>


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It's interesting that neither Marcel nor I could
conjure up a an argument for digits^6.
Maybe there's some fancy complexity argument that
gives this asymptotically. But my finger counting
couldn't get beyond digits^5.
So maybe instead of A*(d+const)^6 one should
just fix A*d^c, at d digits. Including a constant might
have masked a growth slower than digits^6.
I think one should fit the exponent to the data.
Just plot log(time) against log(digits) and
measure the slope of the best fit.
David

I have also investigated these numbers and in particular the case where b=a+1. I
have found these
two large PRP :

1194^1195 + 1195^1194        (3678 digits)
2658^2659 + 2658^2659        (9106 digits)

Henri


[Non-text portions of this message have been removed]

Hello!

OK, I'll prove all remaining prps with less than 1200
digits, i.e.:

8^519+519^8,
20^471+471^20,
5^1036+1036^5,
56^477+477^56,
98^435+435^98,
21^782+782^21,
32^717+717^32,
365^444+444^365,
423^436+436^423,
34^773+773^34.

Best wishes,

Andrey
--------------------------------------------------
13-14 ÉÀÎÑ × ËÌÕÂÅ òÅÁËÔÏÒ ÐÒÏÊÄ£Ô òÅÓÐÕÂÌÉËÁÎÓËÉÊ ÆÅÓÔÉ×ÁÌØ
ÈÕÄÏÖÅÓÔ×ÅÎÎÏÊ ÔÁÔÕÉÒÏ×ËÉ "SNAKE-TATTOO 2001" Ó ÕÞÁÓÔÉÅÍ
ÓÁÌÏÎÏ× íÉÎÓËÁ É âÅÌÁÒÕÓÉ. óÐÒÁ×ËÉ ÐÏ ÔÅÌ. 232-82-51

Let me ask what are the acromyms PRP that was mentioned here and PRF ?

Thanks,noamkj

_________________________________________________________________________
Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com.

Hello,

Has anyone on this list checked out my numbers?
Anyone want to know the theory?
I need help writing the paper(s),
can anyone help me?
Nothing worth writing about here? - I need to
understand why not before wasting my time, or yours.

Thank you

-Dick Boland


--- Dick Boland <richard042@...> wrote:
> Hello,
>
> When g=2762, g^2=7628644,
> My distribution function,
> pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2)
> predicts 350 primes vs. actual 390 primes,
> error= -40 or -10.2564%
>
> I conjecture that g=2762 is the highest g for which the
> deviation error is greater than 10%.
> And I would that someone more skilled than I on this
> list can search for a counterexample.
>
> So far I've tested all g up to g=5250, and I had previously
> tested all prime g up to ~17,000.
>
> As I continue to suspect about this function that
> the percentage of the error deviation grows progressively
> smaller in amplitude, I began testing a range starting
> g=25000 and I would further conjecture that the highest
> g with percentage error > 8% will have occurred
> prior to g=25000.
>
> It was a theoretical scenario that brought me to test this
> function in this neighborhood. I believe my theoretical
> argument will make it clear why this phenomenon must
> exist within the distribution of prime numbers.
>
> > where the constant c = 3/2*ln(3/2)-1/2*ln(1/2) = 0.95477...
>
> Be aware that my first formulation of
>  pi(3*g/2)-pi(g/2) ~ pi(g^2)-pi((g-1)^2)
> may not be the most exact center for this
> "order 1 order 2 codependancy"
> within the distribution of primes, but it is close enough
> that the percentage error goes to zero with increasing g.
>
> I conjecture that one could consider
>
> pi(3*g/2)-pi(g/2) ~ pi((g-1)^2)-pi((g-2)^2) or
> pi(3*g/2)-pi(g/2) ~ pi((g+1)^2)-pi(g^2), for example
>
> and these functions will also yield a percentage error that
> goes to zero, maybe slower, maybe faster, somewhere there may
> be an exact center (error drops fastest).
>
> As I continue to suspect about this function that
> the percentage of the error deviation grows progressively
> smaller with increasing g.
> I began testing a range starting
> g=25000 and now I further conjecture that the highest g
> for which the percentage error exceeds 8% will have occurred
> prior to g=25000.
>
> Here's as far as I got from g=25,000.  The
> highest percentage error found is < 4% in the tests below.
> The sign of the error continues to change frequently
> and the percentage of error continues to average
> lower & lower.
>
> Can someone please verify some of these numbers for me?
>
> Thanks,
>
> -Dick Boland
>
> Data for g>25000
> g      g^2        PRED.  ACT. ERROR count and %deviation
> ______________________________________________________
> 25000  625000000  2476  2431  45  1.8510900863842040312
>  25001  625050001  2477  2475  2  0.080808080808080808
>  25002  625100004  2477  2421  56  2.3130937629078893018
>  25003  625150009  2477  2472  5  0.2022653721682847896
>  25004  625200016  2477  2465  12  0.4868154158215010141
>  25005  625250025  2478  2465  13  0.527383367139959432
>  25006  625300036  2478  2439  39  1.5990159901599015989
>  25007  625350049  2478  2470  8  0.3238866396761133602
>  25008  625400064  2478  2390  88  3.68200836820083682
>  25009  625450081  2478  2503 -25 -0.9988014382740711146
>  25010  625500100  2478  2489 -11 -0.4419445560466050622
>  25011  625550121  2478  2480 -2 -0.0806451612903225806
>  25012  625600144  2479  2466  13  0.5271695052716950526
>  25013  625650169  2479  2497 -18 -0.7208650380456547856
>  25014  625700196  2479  2483 -4 -0.1610954490535642368
>  25015  625750225  2479  2473  6  0.2426202992317023857
>  25016  625800256  2479  2468  11  0.4457050243111831442
>  25017  625850289  2479  2428  51  2.1004942339373970345
>  25018  625900324  2479  2428  51  2.1004942339373970345
>  25019  625950361  2479  2467  12  0.4864207539521686258
>  25020  626000400  2480  2466  14  0.5677210056772100567
>  25021  626050441  2480  2470  10  0.4048582995951417003
>  25022  626100484  2480  2453  27  1.1006930289441500203
>  25023  626150529  2480  2487 -7 -0.2814636107760353839
>  25024  626200576  2479  2493 -14 -0.5615724027276373846
>  25025  626250625  2480  2429  51  2.0996294771510909839
>  25026  626300676  2480  2465  15  0.6085192697768762677
>  25027  626350729  2480  2492 -12 -0.4815409309791332263
>  25028  626400784  2480  2400  80  3.3333333333333333333
>  25029  626450841  2480  2516 -36 -1.4308426073131955484
>  25030  626500900  2480  2512 -32 -1.2738853503184713375
>  25031  626550961  2480  2520 -40 -1.5873015873015873015
>  25032  626601024  2481  2490 -9 -0.3614457831325301204
>  25033  626651089  2482  2471  11  0.4451639012545528126
>  25034  626701156  2482  2486 -4 -0.1609010458567980691
>  25035  626751225  2482  2489 -7 -0.2812374447569304941
>  25036  626801296  2481  2426  55  2.2671063478977741137
>  25037  626851369  2481  2510 -29 -1.1553784860557768923
>  25038  626901444  2481  2448  33  1.3480392156862745097
>  25039  626951521  2481  2456  25  1.0179153094462540716
>  25040  627001600  2481  2469  12  0.4860267314702308626
>  25041  627051681  2482  2486 -4 -0.1609010458567980691
>  25042  627101764  2482  2472  10  0.4045307443365695792
>  25043  627151849  2482  2477  5  0.2018570851836899474
>
> __________________________________________________
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> Get personalized email addresses from Yahoo! Mail - only $35
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I have not been able to find my extended lists of strong pseudoprimes of
the form x^y+y^x.  I'm restarting the search from the largest still
available to me, (1015,384).

More will be posted when reasonable progress has been made.


Paul

Hello!

>I have not been able to find my extended lists of strong pseudoprimes of
>the form x^y+y^x.  I'm restarting the search from the largest still
>available to me, (1015,384).

I've already searched up to y=1227...

Would you start the search from y=1501 please?

Thanks,

Andrey

On Mon, 04 June 2001, "Noam K" wrote:
>
> Let me ask what are the acromyms PRP that was mentioned here and PRF ?


If PRP and PRF are mentioned within a paragraph of each other then
<offtopic topic=cryptography>
PRP = Pseudo-random permutation. i.e an idealised cypher.
PRF = Pseudo-random function. i.e an idealised hash.
</offtopic>

However, to prime number buffs, PRP cvertainly has a meaning, but AFAIK PRF
doesnt.
PRP = PRobable Prime.

A probable prime is a number which has passed a large enough number of
independent probable primality tests, so that it's more likely that you'll be
struck by lightning on the way to collect your million dollar lottery winnings
than it is that the number will be found to be composite. (that's not a strict
definition, but it's a useful one to bear in mind when making bets)

If you perform a test that determinies with no possibility for error that the
number is prime or composite, then you no longer call it a PRP, and it either
becomes
- a prime (woo!)
- a Pseudo-Prime, or PSP

As there are many different probable prime tests, and some can be run with
different parameters, there are different types of PRP and PSP.

All this and more can be found by browsing the marvelous Prime Pages web site. A
good starting point given your question is:
http://www.utm.edu/research/primes/prove/prove2.html

Phil

Mathematics should not have to involve martyrdom;
Support Eric Weisstein, see http://mathworld.wolfram.com
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Hello!

Here's a list of primes and prp's of the form a^b+b^a having less than 2150
digits, 1<a<b:

2^3+3^2, 2 digits, proved by Paul Leyland
2^9+9^2, 3 digits, proved by Paul Leyland
2^15+15^2, 5 digits, proved by Paul Leyland
2^21+21^2, 7digits, proved by Paul Leyland
2^33+33^2, 10 digits, proved by Paul Leyland
5^24+24^5, 17 digits, proved by Paul Leyland
3^56+56^3, 27digits, proved by Paul Leyland
15^32+32^15, 38 digits, proved by Paul Leyland
7^54+54^7, 46 digits, proved by Paul Leyland
33^38+38^33, 58 digits, proved by Paul Leyland
8^69+69^8, 63 digits, proved by Paul Leyland
9^76+76^9, 73 digits, proved by Paul Leyland
21^68+68^21, 90 digits, proved by Paul Leyland
34^75+75^34, 115 digits, proved by Paul Leyland
9^122+122^9, 117 digits, proved by Paul Leyland
56^87+87^56, 153 digits, proved by Paul Leyland
80^81+81^80, 155 digits, proved by Paul Leyland
32^135+135^32, 204 digits, proved by Paul Leyland
67^114+114^67, 209 digits, proved by Paul Leyland
97^114+114^97, 227 digits, proved by Paul Leyland
65^144+144^65, 262 digits, proved by Paul Leyland
45^158+158^45, 262 digits, proved by Paul Leyland
36^185+185^36, 288 digits, proved by Paul Leyland
133^160+160^133, 340 digits, proved by Paul Leyland
98^171+171^98, 341 digits, proved by Paul Leyland
51^206+206^51, 352 digits, proved by Paul Leyland
9^422+422^9, 403 digits, proved by Paul Leyland
76^215+215^76, 405 digits, proved by Paul Leyland
20^357+357^20, 465 digits, proved by Paul Leyland
8^519+519^8, 469 digits, proved by Andrey Kulsha
157^214+214^157, 470 digits, proved by Paul Leyland
87^248+248^87, 482 digits, proved by Paul Leyland
200^237+237^200, 546 digits, proved by Paul Leyland
214^235+235^214, 548 digits, proved by Paul Leyland
20^471+471^20, 613 digits, proved by Andrey Kulsha
91^318+318^91, 623 digits, proved by Paul Leyland
111^322+322^111, 659 digits, proved by Paul Leyland
122^333+333^122, 695 digits, proved by Paul Leyland
5^1036+1036^5, 725 digits, will be proved by Andrey Kulsha
246^318+318^247, 761 digits, proved by Paul Leyland
142^387+387^142, 833 digits, proved by Paul Leyland
56^477+477^56, 834 digits, will be proved by Andrey Kulsha
98^435+435^98, 867 digits, will be proved by Andrey Kulsha
342^343+343^342, 870 digits, proved by Paul Leyland
184^425+425^184, 963 digits, proved by Paul Leyland
289^406+406^289, 1000 digits, proved by Paul Leyland
21^782+782^21, 1034 digits, will be proved by Andrey Kulsha
364^405+405^364, 1038 digits, proved by Paul Leyland
32^717+717^32, 1080 digits, will be proved by Andrey Kulsha
365^444+444^365, 1138 digits, proved by Andrey Kulsha
423^436+436^423, 1146 digits, will be proved by Andrey Kulsha
34^773+773^34, 1184 digits, will be proved by Andrey Kulsha
91^636+636^91, 1246 digits, will be proved by Christ van Willegen
157^580+580^157, 1274 digits, will be proved by Christ van Willegen
329^510+510^329, 1284 digits, will be proved by Christ van Willegen
441^488+488^441, 1291 digits, will be proved by Christ van Willegen
234^545+545^234, 1292 digits, will be proved by Christ van Willegen
234^557+557^234, 1320 digits, will be proved by Christ van Willegen
98^663+663^98, 1321 digits, proved by Christ van Willegen
87^734+734^87, 1424 digits, will be proved by Christ van Willegen
291^590+590^291, 1454 digits, will be proved by Christ van Willegen
513^590+590^513, 1599 digits, will be proved by Christ van Willegen
379^648+648^379, 1671 digits, will be proved by Christ van Willegen
68^927+927^68, 1699 digits, will be proved by Christ van Willegen
54^983+983^54, 1703 digits, will be proved by Christ van Willegen
325^714+714^325, 1794 digits, will be proved by Christ van Willegen
464^675+675^464, 1800 digits, will be proved by Christ van Willegen
191^798+798^191, 1821 digits, will be proved by Christ van Willegen
518^681+681^518, 1849 digits, will be proved by Christ van Willegen
298^815+815^298, 2017 digits, will be proved by Liaskovsky Peter
654^733+733^654, 2064 digits, will be proved by Liaskovsky Peter
61^1156+1156^61, 2064 digits, will be proved by Liaskovsky Peter
95^1044+1044^95, 2065 digits, will be proved by Liaskovsky Peter
443^790+790^443, 2091 digits, will be proved by Liaskovsky Peter
634^747+747^634, 2094 digits, will be proved by Liaskovsky Peter
441^806+806^441, 2132 digits, will be proved by Liaskovsky Peter
45^1298+1298^45, 2146 digits, will be proved by Liaskovsky Peter

Thanks for any comments,

Andrey



[Non-text portions of this message have been removed]

Hello, Paul!

Paul Layland wrote:

> http://research.microsoft.com/~pleyland/primes/xyyx.htm
>
>This page is very much a work in progress and is not linked to from
>anywhere yet.  Please let me know your comments and any additional data
>to be added.   Note that the contents of recent emails haven't yet been
>added.

Good!
It will be yet better if you add "number of digits" column to the table.
Do you have a possibility to add Titanix certificates to your page? I have
all of them (v.2.1) for numbers up to 658 digits.

>In the mean time, I will kill off the searcher and not start it again
>until we're better coordinated.

Let's coordinate thus:
I'll find all SPRPs having no more than 5000 digits, and you'll find all
SPRPs having from 5001 to 10000 digits inclusive. My results will be
obtained within next 10 days.

What machine do you have?

Thanks for comments,

Andrey

Hello!

Dear provers,

please make the detailed report file and send zipped *.in,
*.bpr, *.out and *.cr files named as "5_1036.in" to me.

Great thanks,

Andrey
--------------------------------------------------
13-14 ÉÀÎÑ × ËÌÕÂÅ òÅÁËÔÏÒ ÐÒÏÊÄ£Ô òÅÓÐÕÂÌÉËÁÎÓËÉÊ ÆÅÓÔÉ×ÁÌØ
ÈÕÄÏÖÅÓÔ×ÅÎÎÏÊ ÔÁÔÕÉÒÏ×ËÉ "SNAKE-TATTOO 2001" Ó ÕÞÁÓÔÉÅÍ
ÓÁÌÏÎÏ× íÉÎÓËÁ É âÅÌÁÒÕÓÉ. óÐÒÁ×ËÉ ÐÏ ÔÅÌ. 232-82-51

On Mon, 04 June 2001, "Andrey Kulsha" wrote:
>
> Hello!
>
> Here's a list of primes and prp's of the form a^b+b^a having less than 2150
digits, 1<a<b:
>
> 2^3+3^2, 2 digits, proved by Paul Leyland
> 2^9+9^2, 3 digits, proved by Paul Leyland
[SNIP]

At the primenumbers group pages, in the databases section, there is now a table
called 'a^b+b^a' which represents the above.

The whole group has edit permissions. Be responsible please :-)

If you wish to relieve yourself of responsibility, just blank your name out, and
if you have proved a PRP to be prime just change that column. If you find a
pseudo-prime, can I recommend you mark it as such, and let the list know?

It was trivial to set up such a table, and the feature is great for recording
the state of these kinds of friendly free-for-alls.
Feel free to ask me for more information if you think there's another table that
might be useful for such sharing.

Phil

Mathematics should not have to involve martyrdom;
Support Eric Weisstein, see http://mathworld.wolfram.com
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On Mon, 04 June 2001, Dick Boland wrote:
>
> Hello,
>
> Has anyone on this list checked out my numbers?

We probably all trust you to have got the numerics correct, so 'checked' may not
be the right word. They certainly look believable.

> Anyone want to know the theory?
> I need help writing the paper(s),
> can anyone help me?
> Nothing worth writing about here? - I need to
> understand why not before wasting my time, or yours.

You need more data, from far higher ranges, before such a prediction makes much
sense. When n is small the read deviation may be smaller than the noise.

If you look at www.wolfram.com (the Mathematica website), then I know in the
'Mathematica Book' section, there's am implementation note:
<<<
Prime and PrimePi use sparse caching and sieving. For large n, the
Lagarias­Miller­Odlyzko algorithm for PrimePi is
used, based on asymptotic estimates of the density of primes, and is inverted to
give Prime.
>>>

Using those names you could try to find the algorithm in question, and using
that find some far higher ranges to prove (in the original sense, meaning
'test') your hypothesis.

You might be able to find an online calculator, or Java Applet which does the
calculation for you. ('Prime Pi' is the standard name for the function, so it
probably a good search string.)

Good luck,
Phil

Mathematics should not have to involve martyrdom;
Support Eric Weisstein, see http://mathworld.wolfram.com
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Hello!

Phil Carmody wrote:

> At the primenumbers group pages, in the databases section, there is now a
table called 'a^b+b^a' which represents the above.

I've added the "finder" column to the table.

It needs to be filled, but I don't know how to do it. There's an info:

Finder = Henri Lifchitz:
a=1194, b=1195;
a=2658, b=2659.

Finder = Andrey Kulsha:
a=45, b=1298;
a=2, b=2007;
a=2, b=2127;
a=2, b=3759.

Finder=Paul Leyland:
all remaining numbers.

Note that the column "prover" means who is a prover (if status = prime) or
will be a prover (if status = PRP).

Is it possible to make the table sorted by "digits" as default?

Thanks,

Andrey

Hello again,

and the last question: how to change only one cell of a table?

Thanks,

Andrey



[Non-text portions of this message have been removed]

Phil Carmody wrote:
> You might be able to find an online calculator
http://www.math.Princeton.EDU/~arbooker/nthprime.html

Can someone please post a simple guide to restarting a primality test
with Titanix V2.1 i.e. after aborting what input do you chose?
I'd also like to know if there's any relationship between the index
number and its chance of reducing the number of bits or with how many
bits are removed.

I've decided to have a go at one of my numbers, phi(5,244!) (1913d)

Andrew Walker

Andrew Walker asked:

> after aborting what input do you chose?

The previous R is the next N.
If there was a break, you have to restart
by inputting the last R manually.

Then, at the end, paste the bits of the certs
together, and use Jim's

Cert_Val -quick_check

to check that your combined cert parses OK.
Then run Cert_Val, for real, to get validation
(much quicker than obtaining the cert)

Good luck with

> phi(5,244!) (1913d)

David

Conjecture: The average number of bits in N/R,
whittled away at one step of ECPP,
is independent of N.

Note: It may very well depend on the implementation!

If this be supported by the data, then there is
a better way to combine them than has been
done heretofore. Namely take all the steps from the
the magnificent La Barbera record, and plot
log(step_time) against log(N).

Of course, there will be a lot of scatter.
Nevertheless, let s be the slope of the best fit,
in the region from 2k to 4k digits.

Then if the conjecture is reasonable, ECPP,
in the fitted region, scales (empirically)
like digits^(s+1).

I have the feeling that s < 5.

David

Hi,

I checked the conjecture by using the "nthprime"
page which David  Broadhurst proposed and found
for
g=10^6, that
pi(g^2)-pi((g-1)^2)= 72470 while
pi(3*g/2)-pi(g/2)  = 72617
with a relative difference of 3.4e-3.
Thus, it seems working pretty well. An exact
proof would be most  interesting, especially if
providing error bounds.

Ferenc
2,3,5,7,17,23,47,103,107,137,283,313,347,373,...

Hello,

I'm looking for a good basic book on prime numbers and
number theory, preferaby with a strong Riemann slant.
The only book available from Amazon.com is 215$US. Any
suggestions? Alternatively or in the interim, could
anyone dummy up Riemann's theory enough for a layman
to comprehend?

Bill

=====
Bill Krys
Email: billkrys@...
Toronto, Canada (currently: Beijing, China)

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pi(x) ~ x/ln(x)*(1+1/ln(x)+O(1/ln(x)^2))
  lhs = pi(g^2)-pi((g-1)^2)
  rhs = pi(3*g/2)-p(g/2)
  rhs/lhs = 1 + k/log(g) + O(1/ln(g)^2)
  k = 1 - log(27/4)/2 = 0.04522874755778077232...

Hence rhs > lhs, at large g, because the
base of Naperian logarithms exceeds sqrt(27/4).

> In Crandall/Pomerance "Prime Numbers" on the pages 381-382
> you can find a longish polynomial. The last pair of parenthesis
> on page 381 seem not to be necessary, i.e. ...-(x+c*u^2)...
> could be written shorter as -x-c*u^2, unless there is a typo
> somewhere.
>
> Has someone checked this polynomial?

Congratulations to Hans for spotting the first error in C&P. It ought to be
"-(x+cu)^2"


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Let

L(g) = pi(g^2) - pi((g-1)^2)
R(g) = pi(3*g/2) - pi(g/2)
D(g) = R(g) - L(g)

where pi(g) is the number of primes not exceeding g.

Dick Boland conjectured that D(g) changes
sign an infinite number of times.

On the contrary, I claimed that

k = lim_{g to infty} log(g)^2*D(g)/g = 1 - log(27/4)/2 > 0.

If you replace pi(x) by Riemann's estimator R(x)
(Ribenboim p224) you will find a single sign change
around g=10^4. Superimposed on this upward trend
are sqrt fluctuations from the complex zeros of zeta.
Dick was misled by the fact these can easily buck
the trend for his small g's, around 2.5*10^4.

But for how much longer can this go on?

Already it's getting difficult for g around 10^6,
where a simple sieve of Eratosthenes gave

    g     R(g)  L(g) D(g)
1000000 72617 72450 167  [Pace Ferenc]
  999999 72617 72569  48
  999998 72617 72340 277
  999997 72617 72573  44
  999996 72617 72546  71
  999995 72617 72381 236
  999994 72617 72542  75
  999993 72617 72425 192
  999992 72617 72548  69
  999991 72617 72180 437
  999990 72617 72195 422
  999989 72617 72561  56
  999988 72617 72434 183
  999987 72617 72703 -86  [Made it!]
  999986 72617 72099 518
  999985 72617 72162 455
  999984 72616 72378 238
  999983 72616 72317 299
  999982 72616 72511 105
  999981 72616 72371 245
  999980 72616 72579  37
  999979 72616 72311 305
  999978 72616 72352 264
  999977 72616 72548  68
  999976 72616 72645 -29 [And again!]

These *roughly* agree with a mean k*g/log(g)^2 = 237
and a deviation that is of order sqrt(g/log(g))= 269.

Puzzle: Is there a g>10^7 for which D(g)<0 ?

Here it won't be so easy to
buck the Riemann trend, since
(k*g/log(g)^2)/sqrt(g/log(g)) > 1741/788 >  2.2

Hello,

> Prime and PrimePi use sparse caching and sieving. For large n, the
Lagarias­Miller­Odlyzko
> algorithm for PrimePi is
> used, based on asymptotic estimates of the density of primes, and is inverted
to give Prime.

  Thanks Phil,
  Interesting stuff rersulting from this search (besides the algorithm),
  I will be doing some research to try and put it into context of my theory
  I haven't gotten my hands on the algorithm in a form that I can use,
  and it would be good to get some higher data, but it may not be necessary.
  The highest prime page is good for some spot checking as Forenc showed,
  and still no counterexamples :)

  As for Dave's proposition
> pi(x) ~ x/ln(x)*(1+1/ln(x)+O(1/ln(x)^2))
> lhs = pi(g^2)-pi((g-1)^2)
> rhs = pi(3*g/2)-p(g/2)
> rhs/lhs = 1 + k/log(g) + O(1/ln(g)^2)
> k = 1 - log(27/4)/2 = 0.04522874755778077232...
> Hence rhs > lhs, at large g, because the
> base of Naperian logarithms exceeds sqrt(27/4).

I'm not sure that the above proves anything
or if it simply reflects what current
wisdom on the subject would have us believe.
If it's a hard mathematical proof, it would seem to disprove
the conjecture that the sign of the error in my function
changes infinitely often, but not necessarily disprove the
percentage error going to zero.
I need to understand it better, so I have some home work.

I should be able to put something together to share after the weekend.

Thank you,

-Dick Boland



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Hello!

There are some news/problems concerned pi(x) function.

Some weeks ago I've found a good approximation for an amplitude of
fluctuations of pi(x).

Let pi(x)=P(x)+A(x), where P(x) is a "mean", or "smooth" part of pi(x),
and A(x) is a remainder. Then my conjecture is:

|A(x)|=O[sqrt(x*(loglogx)^(1+eps))/logx] for some eps>0.

Of course, it hasn't yet proven, but heuristically it's true. Namely,
the probability of the falsehood aims to 0 when x goes to +infinity.
I have yet stronger heuristic result for the mean value of |A(x)| around
some x:

<|A(x)|>=C1*sqrt(x*(loglogx+O[1]))/(logx-C2),
where C1, C2 are positive constants.

But, unfortunately, I can't find a way to prove these results.
I know that this way (not a proof, only way!) is maybe shown in some
papers concerning fluctuations of M(n)=sum(mu(k),k,1,n); some references
we can find in Andrew Odlyzko's paper "Disproof of Merten's conjecture"
http://www.research.att.com/~amo/doc/arch/mertens.disproof.pdf ,
but I have no access to those references. :-(

* * *

As for P(x). I haven't yet obtained any useful results, but perhaps
there's the way to find them.

Let's transform the usual representation of Riemann's estimator

R(x)=sum(mu(n)*Li(x^(1/n))/n,n,1,+infinity)

using the series

Li(x)=Euler+loglogx+sum((logx)^k/k!/k,k,1,+infinity),
where Euler=0.57721...;

we know that

sum(mu(k)/k^n,k,1,+infinity)=1/zeta(n),
sum(mu(k)*ln(k)/k,k,1,+infinity)=-1,

so the result is:

R(x)=1+sum((logx)^n/n!/n/zeta(n+1),n,1,+infinity).

There is a way to find P(x):

P(x)=1+sum((logx)^n/n!/n/func(n+1),n,1,+infinity).

Here "func" is a function close to zeta, it goes to 1+ when n goes to
+infinity.

We see: at first Gauss conjectured that func=1, i.e. P(x)~Li(x). But
this was proven to be wrong.

Then Riemann conjectured func=zeta, and now we know it's wrong too (a
contradiction with Littlewood's result).

I still have no any ideas about func(n+1).

Maybe somebody know about any papers/books discussing this problems?

Great thanks,

Andrey

The Riemann Conjecture has not been proved wrong yet.
The Clay Institute are still looking after the $1,000,000 for me ;-)

I do remember reading that someone had proved that a Littlewood
Conjecture _OR_ Riemann's Hypothesis is FALSE.
I think this must have been on the web and I have half a memory that
the title misused the word "paradox".
I have tried searching the web and my browser's history but I can't
find it again :-(

Does anyone know the reference or can suggest anything to tighten
my searches?

Cheers,
Paul Landon

Andrey Kulsha wrote:
[snip]

> We see: at first Gauss conjectured that func=1, i.e. P(x)~Li(x). But
> this was proven to be wrong.
>
> Then Riemann conjectured func=zeta, and now we know it's wrong too (a
> contradiction with Littlewood's result).
>
> I still have no any ideas about func(n+1).
>
> Maybe somebody know about any papers/books discussing this problems?
>
> Great thanks,
>
> Andrey
>
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