FAQ: Where is a list of the x-digit primes?

By Chris Caldwell

Question: Where can I find a complete list of the 106-digit primes?

Answer: "Shouldn't be hard to find. In every one of the complete lists of 106-digit numbers that's ever been published, the primes are clearly marked" [ Dave Rusin, on sci.math]

If you did not find that answer funny, then you definitely need to read the rest of this entry! 

How many 106-digit number are there?  Well, the smallest is 10105 and the largest is 10106-1, so there are 10106-10105 of them.  There are nowhere this many atoms in the entire universe, so of course there is no such list of integers.

What if we just want the primes?  That does not help much.  By the Prime Number Theorem the number of primes less than x is about  x/log x  where log x is the natural logarithm of x (roughly 2.3 times the number of digits in x). So the number of 106-digit primes is about

39086503 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000 0000000000.

Good luck fitting that list in this universe!

So how big can x be and we still have a complete list? That depends on how much of the world's resources you want to dedicate to the list.  If you want to stick with a single computer, then definitely less than 20 digits. You might, with proper compression techniques, create a list of all 16-digit primes.

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