Thread: Who has a list?
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Old 2003-12-22, 20:41   #3
ewmayer
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Sep 2002
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Any list of the actual primes below some bound B grows exponentially fast with the number of digits in B (i.e. with log B, where the log is to any base), so such lists are not generally kept, as it is faster to simply generate said primes when needed and only explicitly store the ones that are needed at the moment. We know that the number of primes < B scales as B/ln B (natural log here), so e.g. below 2^32 (rouhgly 10^10) we have nearly 2^28 primes, each of which would need ~4 bytes to store explicitly, for a total of roughly a Gigabyte. There are ~10^98 primes < 10^100, which is vastly more than the number of elemntary particles in the known universe. What people have done along these lines is to actually count the number of primes below some ever-increasing bound, to check the accuracy of various asymptotic formulae which approximate the prime-count function. At present, the prime count has been exactly enumerated to slightly above 10^20, so 10^100 is a long way off! Remember, actual enumeration requires on the order of as many CPU-cycles as primes, and in all of human history there have been just slightly above a mole (6.23...X 10^23) of CPU-cycles. Even were this count to double each year, it would take over 300 years to reach 10^100.

Here are some related links from MathWorld:

http://mathworld.wolfram.com/PrimeNumberTheorem.html

http://mathworld.wolfram.com/PrimeCountingFunction.html
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