Silverman & Wagstaff on Joint Distribution of Ultimate and Penultimate Prime Factors

[wblipp]

Originally Posted by wblipp

This thread pulls together discussions that have been buried on several different threads.

There has recently been renewed discussion about Silverman and Wagstaff's paper A Practical Analysis of Elliptical Curve Factoring.

In this message we have agreement that Brent's version of the 2-parameter Dickman function is correct, along with an assurance that the error is cosmetic:

http://www.mersenneforum.org/showpos...0&postcount=68


In this message we have a summary of the idea as using Baye's Theorem to update the Dickman function:

http://www.mersenneforum.org/showpos...7&postcount=15


At first glance it appears that changing the Dickman function should change Equation 4.6, but I haven't yet taken the time to look into that issue.

[R.D. Silverman]

Originally Posted by wblipp

In this message we have a summary of the idea as using Baye's Theorem to update the Dickman function:

Actually, the Bayes' Thm. computations in the paper use the following theorem for the prior, rather than Dickman's function:

The probability that a large integer X has a factor between y and y^(1+e)
is e/(e+1). So, for example, the probability that an integer (say) near
200 digits has a factor between 50 and 60 digits gives e = .2 and a
probability of 1/6. This allows use to compute a density function for
the probability that a factor exists in the range of interest. We then apply
Bayes' Thm. using the ECM failures as a sample.