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carpetpool 2019-08-20 03:02
Probability N has a prime factor > sqrt(N)
For some integer N chosen at random, what is the probability that N has a prime factor > sqrt(N)? This also includes when N is prime, therefore the probability is greater than 1/ln(N).
Furthermore, what's the probability that N has a prime factor > N^(1/k) ?
Thanks for any new leads.
wblipp 2019-08-21 01:33
[QUOTE=carpetpool;523989]For some integer N chosen at random, what is the probability that N has a prime factor > sqrt(N)?[/QUOTE]
See Dickman–de Bruijn function in [URL="https://en.wikipedia.org/wiki/Dickman_function"]Wikipedia[/URL], [URL="http://mathworld.wolfram.com/DickmanFunction.html"]Wolfram[/URL] and elsewhere.
fivemack 2019-08-21 10:16
It takes a long time to reach an asymptote ...
c=0;for(t=10^12,10^12+10^6,F=factor(t);lp=F[matsize(F)[1],1];if(lp*lp<=t,c=1+c)); c
The normal suggestion is that it's about k^-k, so 1/4. But sampling ranges of 10^6 at different places suggests that the count (and so the implied probability) goes up perceptibly for larger N
1e6 270639
1e10 282202
Assuming that we're dealing with a Poisson process with a fixed probability at each sampling point, which I don't think is unfair, those observations are not consistent with the probability being the same at 1e6 as at 1e18.
R. Gerbicz 2019-08-21 10:29
[QUOTE=fivemack;524128][code]
For k=2 the exact result is known, it is 1-log(2).