Go Back > Math Stuff > Other Mathematical Topics

Thread Tools
Old 2011-03-06, 00:42   #12
mart_r's Avatar
Dec 2008
you know...around...

22·5·29 Posts

Slightly modified to make it sound more sophisticated :

Originally Posted by mart_r View Post
Let q(n,p) denote the parameter such that the actual probability of primality for a number n, when trial divided up to p, equals \frac{\log n}{q(n,p)*\log p}.
Applying SOE for an appropriate range in the vicinity of a given n and juxtaposing the values for n = \infty, the values q(n,p) for the fixed n fluctuate around the other, q(\infty,p), as p increases (...)
I wrote "actual probability" in contrast to the "theoretical probability" which is q(\infty,p)
The "appropriate range" should not be too small (<~n1/4) to avoid a ragged graph, but not too big either (>~n/10) to keep n "within the focus".

I'll put this into another variable: r(n,p) = q(n,p)-q(\infty,p).

For 1<p<n^{(1/e)}, r(n,p) has a
- local minimum of about -0.018 at p\approx n^{0.362} and a
- local maximum of about 0.0021 at p\approx n^{0.288}. Furthermore, for 1<p<n^{0.288}, a
- local minimum of about -0.00014 at p\approx n^{0.237}, preceded by a
- local maximum of about 0.00003 at p\approx n^{0.202} in the range 1<p<n^{0.237}.
Enough yet? The preceding minimum should be somewhere around p\approx n^{0.164}.

These are numbers taken from 109 sample values between 1016-5*108 and 1016+5*108 and might not be accurate to the last digit if this pattern continues that way for n --> \infty.

I still can't make head nor tail of this phenomenon, but I would guess these numbers are connected to something which is already known, i.e. can be calculated from well-known constants. But for now, I've got to get some sleep.
mart_r is offline   Reply With Quote
Old 2020-07-20, 13:37   #13
mart_r's Avatar
Dec 2008
you know...around...

22·5·29 Posts

If there's a single person other than me out there who was stumped by this problem, let me solve it by referring to the Buchstab function. This paper, on page 11, contains the very answers to my question:
(A. Y. Cheer and D. A. Goldston: A differential delay equation arising from the sieve of Eratosthenes)

- local minimum at p ~ n^(1/c2) = n^0.3618962566...
- local maximum p ~ n^(1/c3) = n^0.288206...
and so on and so forth.

If it was a message to myself in the past, I'd also suggest looking into the parity problem.

P.S.: sorry for the gravedigging, but I had to post it here, I remember that I linked to this thread at least on one occasion.

Last fiddled with by mart_r on 2020-07-20 at 13:43 Reason: + link description
mart_r is offline   Reply With Quote

Thread Tools

Similar Threads
Thread Thread Starter Forum Replies Last Post
probability ATH Homework Help 7 2014-10-23 00:50
Probability of factor (TF) nuggetprime Math 2 2011-03-19 22:14
Primality searches and primality successes marco_calabresi Information & Answers 3 2009-04-17 19:44
P-1 Probability question JuanTutors Factoring 2 2005-01-12 20:41
What is the probability distribution for M42 ? dsouza123 Math 2 2004-06-02 02:16

All times are UTC. The time now is 19:52.

Wed Sep 30 19:52:23 UTC 2020 up 20 days, 17:03, 0 users, load averages: 1.79, 1.93, 1.83

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.