mersenneforum.org > Math Probability of finding a factor in TF
 Register FAQ Search Today's Posts Mark Forums Read

2003-04-28, 19:51   #1
eepiccolo

Dec 2002
Frederick County, MD

2·5·37 Posts
Probability of finding a factor in TF

Alrighty,

At http://www.mersenne.org/math.htm, we are told:
Quote:
 the chance of finding a factor between 2^x and 2^x+1 is about 1/x.
My question is this: is this probability dependent or independent of whether or not other factors have been found? And for what values of x is this valid? It doesn't seem like it shouild work for small values of x.

Tim

 2003-04-29, 15:07 #2 eepiccolo     Dec 2002 Frederick County, MD 2·5·37 Posts In a posibly related note, is the following infinite series equal to 1? [code:1]inf Σ 1/(n(n+1)) = 1 ? n=1[/code:1]
 2003-04-29, 19:47 #3 S80780   Jan 2003 far from M40 53 Posts Yes, it is. 1/(n*(n+1)) = (n+1-n)/(n*(n+1)) = (n + 1)/(n*(n+1)) -n/(n*(n+1)) = 1/n - 1/(n+1) So, the kth partial - sum has the value 1 - 1/(k+1) which's limit for k to infinity is 1 q.e.d. Sums like these, where a summand is (partly) nihilized by its successor are called telescope sums. Benjamin
2003-05-09, 02:53   #4

"Richard B. Woods"
Aug 2002
Wisconsin USA

22·3·641 Posts
Re: Probability of finding a factor in TF

Quote:
Originally Posted by eepiccolo
At http://www.mersenne.org/math.htm, we are told:
Quote:
 the chance of finding a factor between 2^x and 2^x+1 is about 1/x.
First, let's note that the sentence begins with "Looking at past factoring data we see that the chance ..." Right now I don't recall whether I've seen a proof of that estimate. Judging by the sentence's wording, it may be an empirical observation.

Quote:
 My question is this: is this probability dependent or independent of whether or not other factors have been found?
If I understand correctly, what is meant is "... the chance that 2^p-1 has a factor between 2^x and 2^(x+1) is approximately 1/x."

That is, it refers to the probability of the existence of a factor (known or unknown) of 2^p-1 within the specified range, not necessarily the probability of discovering a previously-unknown factor there after some factors of 2^p-1 are already known.

 2003-06-07, 05:56 #5 cheesehead     "Richard B. Woods" Aug 2002 Wisconsin USA 22×3×641 Posts For a thorough earlier discussion, see the thread "Does the LL test:s factorization save or waste CPU time?" in The Software forum at http://www.mersenneforum.org/viewtopic.php?t=78, especially svempasnake's table comparing the 1/n prediction to the actual number of factors found, on page 3 at Mon Sep 16, 2002 9:32 pm (http://www.mersenneforum.org/viewtop...highlight=#895).

 Similar Threads Thread Thread Starter Forum Replies Last Post nuggetprime Math 2 2011-03-19 22:14 JuanTutors Software 20 2004-09-26 09:47 optim Math 2 2003-12-06 19:03 Deamiter Math 4 2002-12-25 06:06 Deamiter Software 4 2002-10-11 16:36

All times are UTC. The time now is 12:51.

Sat Sep 26 12:51:01 UTC 2020 up 16 days, 10:01, 1 user, load averages: 3.03, 2.30, 1.93