Hi,

last night I was musing this topic over a few beers, and came up with this

illustration of the 'inaccessability' of large primes (say 100 digits to create

200 digit products). Any mathematicians Finish reading here!

One of the clasic ways of discovering primes is the 'Sieve of Erastothenes'. For

those who don't know it,this works quite well on a computer - you allocate one

bit for each integer in the range you choose. Set them all to 0, integer 1 is a

special case, 2 is 'known' to be a prime so set on every 2nd bit [1/2 of the

candidate set] (because multiples of a prime are by definition non-prime), three

is also a 'known' prime so set on every third bit [1/2 are already set so this

sets on 1/2 * 1/3 = 1/6 of the candidate set], 4 is exactly divisible by one of

the discoverd primes so is non-prime, 5 is prime so set off every 5th bit [2/6 *

1/5 = 1/15 of the candidate set]... and so on ..... One thing this shows is that

primes can be expected to become rarer as the integers increase in size - but

only very slowly and at a decreasing rate (approx. 25% primes in 100; 17% in

1,000; 12% in 10,000; 8% in a million...). So there will be plenty of primes in

the 100 digit range, but how to find them? This (at last <g>) is my

illustration:

If we use 'the Sieve' we will need somewhere to store our bit string. Now 10^100

bits is (say) 10^99 bytes. Suppose we give everyone in the world one or two

large (e.g. 100 gig) disk drives - we would have approx. 10^10 * 10^11 bytes

available = 10^21 of the 10^99 we need. So we need some more planets - about

10^78 of them. If every star in the universe had 10 planets, I don't think we'd

have anywhere near enough!

Cheers (_)? AndyD 8-)#

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