Highest prime number calculated on a regular PC
You don't use a sieve to find cryptographic-size primes.

One method chooses an odd number of the appropriate size at random, then tests it using a Miller-Rabin or Lucas pseudoprime test, incrementing the number by 2 until a probable prime is found; an alternative is to pre-sieve with small primes (less than 1000, say), and perform pseudoprime tests only on those numbers that survive the sieve. That works, and is commonly used in cryptographic practice, but is not guaranteed to find a prime number. An implementation of this method is available at my blog.

Also at my blog is an explanation of a better method for generating cryptographic primes:


  Sometimes you need to have a large prime, for testing, cryptography,
  or some other purpose. I’m talking about primes of several hundred to
  a few thousand digits that are certified &mdash; proven to be prime
  &mdash; rather than probable primes according to a Miller-Rabin or
  other probabilistic test. Henry Pocklington’s Criterion, which dates
  to 1914, gives us a way to find such primes quickly. Paulo Ribenboim
  explains it thus:
  
  Let p be an odd prime, and let k be a positive integer, such that
  p does not divide k and 1 &lt; k &lt; 2(p + 1). Let N = 2kp + 1. Then the following conditions are equivalent:
  
  
  N is a prime.
  There exists an integer a, 1 &lt; a &lt; N, such that a(N&minus;1)/2 &equiv; −1 (mod N) and gcd(ak + 1, N) = 1.
  
  
  This gives us an algorithm for generating large certified primes.
  Choose p a certified prime. Choose 1 &le; k &lt; 2p at
  random; we ignore the last two possibilities for k, so we don’t have
  to worry about k being a multiple of p. Compute N. For each a
  &isin; {2, 3}, test the conditions for primality. If you don’t find a
  prime, go back and choose a different random k. Once you have a
  prime N, “ratchet up” and restart the process with the new certified
  prime N as the p of the next step. Continue until N is big
  enough.


There is code and a more complete explanation at my blog.