Finding primes & proving primality
4.3: Conclusion and suggestions
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In practice it is easy to decide what method of primality proof to use: 

When programming the classical methods, the most difficult aspect is multiplying quickly.  Fortunately someone has done much of the work for us!  There are several large free libraries for arithmetic with large integers as well as for proving the primality of large integers.

With the classical methods you can easily handle a 100,000 digit number.  With the modern methods you will work very hard to handle a 5,000 digit number!  Unless you are very brave I would suggest you look for an already coded version of the modern algorithms, they are quite difficult to implement.

At this site we keep a list of the 5000 largest known primes, so if you do find new record primes, why not let us know

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