mersenneforum.org Odds that a random number is prime
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 2009-08-27, 19:30 #1 Number theory   2×3,761 Posts Odds that a random number is prime A random 321,000 (decimal) digit number is chosen. What is the probability that it's prime if it... a.) Is an odd number? b.) Doesn't have any factors below one billion (10^9)? c.) Doesn't have any factors below one trillion (10^12)? For part A, I got 1 in 370,000 by taking ln(10^321000) and dividing it by 2. Is this right? Also, could you help me out with parts B and C?
 2009-08-28, 00:26 #2 wblipp     "William" May 2003 New Haven 2,371 Posts http://en.wikipedia.org/wiki/Mertens%27_theorems The third theorem is usually rearranged to get an estimate. Last fiddled with by wblipp on 2009-08-28 at 00:26
2009-08-28, 11:26   #3
R.D. Silverman

Nov 2003

22×5×373 Posts

Quote:
 Originally Posted by Number theory A random 321,000 (decimal) digit number is chosen. What is the probability that it's prime if it... a.) Is an odd number? b.) Doesn't have any factors below one billion (10^9)? c.) Doesn't have any factors below one trillion (10^12)? For part A, I got 1 in 370,000 by taking ln(10^321000) and dividing it by 2. Is this right? Also, could you help me out with parts B and C?
(a) 2* [pi(10^321000) - pi(10^320999)]/(10^321000 - 10^320999)

(b) approx. exp(gamma)/[9 * log(10)]

(c) approx. exp(gamma)/[12*log(10)]

Where pi is the prime counting function and gamma is Euler's constant.
Look up Mertens' Theorem.

 2009-08-28, 22:04 #5 Number theory   7×17×61 Posts Thanks.

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