Probability of finding a factor in TF
Alrighty,
At [url]http://www.mersenne.org/math.htm[/url], we are told: [quote]the chance of finding a factor between 2^x and 2^x+1 is about 1/x.[/quote] 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 
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] 
Yes, it is.
1/(n*(n+1)) = (n+1n)/(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 
Re: Probability of finding a factor in TF
[quote="eepiccolo"]At [url]http://www.mersenne.org/math.htm[/url], we are told:
[quote]the chance of finding a factor between 2^x and 2^x+1 is about 1/x.[/quote][/quote] 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?[/quote] If I understand correctly, what is meant is "... the chance that 2^p1 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^p1 within the specified range, not necessarily the probability of discovering a previouslyunknown factor there after some factors of 2^p1 are already known. 
For a thorough earlier discussion, see the thread "Does the LL test:s factorization save or waste CPU time?" in The Software forum at [url=http://www.mersenneforum.org/viewtopic.php?t=78]http://www.mersenneforum.org/viewtopic.php?t=78[/url], 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 ([url=http://www.mersenneforum.org/viewtopic.php?p=895&highlight=#895]http://www.mersenneforum.org/viewtopic.php?p=895&highlight=#895[/url]).

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