random a 512-bit integer N that is not a multiple of 2, 3, or 5

Question

if you are to choose a random a 512-bit integer N that is not a multiple of 2, 3, or 5 What is the probability that N is prime? i don't know the algorithm behind this one... i'm trying to work on a project but this is the starting point.. :)

Answer 1

The number of primes less than n=2⁵¹² is approximately n/log(n). The number of numbers you are considering is 4/15*n, so the probability you are looking for is 15/(4*log(n)), which is about 1 %.

Answer 2

Probability bounds

You may use the following inequality for the prime pi function:

(Where log is taken in base e)

So:

8.58774*10¹⁵¹ < π(2⁵¹²) < 8.93096*10¹⁵¹

And as you are only leaving alive 4/15 n numbers (because of killing he multiples of 2, 3 and 5), te probability is bounded by:

8.58774*10¹⁵¹/(4/15 2⁵¹²) < P < 8.93096*10¹⁵¹/(4/15 2⁵¹²)

Or:

0.010507 < P < 0.010687

Which is a nice, pretty tight bound.