2013-04-25, 11:47 | #1 |
"Lucan"
Dec 2006
England
2×3×13×83 Posts |
Rigour Requested
Probability of a factor between x and x+dx is dx/(xlnx)
Let P(x) be the probability of no factor <x. Then P(x+dx) = P(x)(1-dx/(xlnx)) dP/P = -dx/(xlnx) Integrating from x=a^{X} to x=a^{Y} gives the probability of no factors in this range = X/Y, from where we get the familiar probability of one or more factors between 2^{X-1} and 2^{X} = 1/X. I am grateful to Oliver for the following sanity check: The product of all prime factors of N is N. The sum of their logs is logN. Expected sum of their logs is the integral of logx(1/xlnx)dx from 1 to N = logN. ****** All such simple stuff that there must be some truth in it. However there are two large elephants in the room named "independence" and "2kp+1". Can someone get rid of them please? David Last fiddled with by davieddy on 2013-04-25 at 11:59 |
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