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Old 2004-09-01, 16:56   #3
R.D. Silverman
 
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Nov 2003

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Quote:
Originally Posted by wblipp
Let's take a simple case where we know the answer. Consider the second largest prime is less then than x1/2 and the largest prime factor is less than x2/3.

Since the second largest prime can NEVER exceed x1/2, we know this is the same as the single condition that the largest prime factor is less than x2/3. And we know that this is rho(1.5). And we know that this is 1-ln(1.5) = 0.595.

For using the expression in the paper, we have alpha = 2 and beta = 1.333.

Over these ranges we know rho analytically as 1 or 1-ln(t), so it's easy to evaluate both expressions.

The expression in the paper evaluates to 0.496. The expression in my guess evaluates to 0.595.
The probability that the second largest is less than x1/B is not
independent of the probability that the largest is less than x1/A.
The distribution of the second largest factor depends on the size of the
largest one... If the largest is near x2/3 then the second
largest must be less than x1/3..... You have to consider the
joint pdf and they are not independent. Might this be the source of the
problem? You can't just compute the probability that (say) the largest
is less than x^k and the probability that the second largest is less than
x^j and conclude that the probability that the largest is less than x^k
AND that the second largest is less than x^j is the product of the two...

Dan Bernstein has code that will do these computations. Perhaps you
might want to compare what his code yields with yours...

I am NOT saying that you are wrong. I could very well be in error.
But others have read this paper as well without finding what might be
this problem...
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