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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 10^{105} and the largest is 10^{106}-1, so there are 10^{106}-10^{105} 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.