How do we keep finding primes?

The largest known primes are found on ordinary PC's with free software. Small primes can be computed faster than they can be read from a hard disk, so nobody bothers to store all the primes a PC can compute in a single day. The prime number theorem says the number of primes below x is approximately x/ln(x), whew ln is the natural logarithm. The Goldbach conjecture verification project has computed all primes below 36*10^17 without storing them. That is 86326326584778317 primes. There are free programs which can find thousands of larger and never previously computed primes in a second on a PC. There is only interest in new primes if they are very large or have special properties. There is a website with the 5000 largest known primes. Number 5000 currently has 221567 digits.

This will answer most of your questions: http://en.wikipedia.org/wiki/Great_Internet_Mersenne_Prime_Search GIMPS is a large scale distributed computing project which uses the PCs of many volunteers to run prime-searching algorithms. Not all primes less than the largest are known. Sometimes GIMPS finds smaller ones than the previously found largest. Two smaller ones were found since the largest one, which you gave in your question. There's a link in the article above to a description of the algorithm used by GIMPS. It's http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test This test is optimized for finding Mersenne primes, not primes in general. Finding large primes is rather difficult, so the effort is directed at those that are somewhat easier to find. It's still difficult, but they've found about 1 additional prime per year. Things seemed to have slowed down, since the last was found in 2009. To join the search, go to http://www.mersenne.org/freesoft/#newusers to get the software. I don't know much about it, but I imagine that it handles numbers with a huge number of digits through special handling to avoid the limits built into AMD or Intel or other chips, and Java or the usual calculation limits. That's not that difficult. I wrote a VBA routine to calculate 100!. I just assign a variable to each digit, and do the multiplying digit-by-digit. Something of a pain, but it does easily get around the 15 or so digit limits of Excel. There are other software packages you can get to handle large number arithmetic, but I haven't had to resort to those yet.

This will answer most of your questions: http://en.wikipedia.org/wiki/Great_Internet_Mersenne_Prime_Search GIMPS is a large scale distributed computing project which uses the PCs of many volunteers to run prime-searching algorithms. Not all primes less than the largest are known. Sometimes GIMPS finds smaller ones than the previously found largest. Two smaller ones were found since the largest one, which you gave in your question. There's a link in the article above to a description of the algorithm used by GIMPS. It's http://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test This test is optimized for finding Mersenne primes, not primes in general. Finding large primes is rather difficult, so the effort is directed at those that are somewhat easier to find. It's still difficult, but they've found about 1 additional prime per year. Things seemed to have slowed down, since the last was found in 2009. To join the search, go to http://www.mersenne.org/freesoft/#newusers to get the software. I don't know much about it, but I imagine that it handles numbers with a huge number of digits through special handling to avoid the limits built into AMD or Intel or other chips, and Java or the usual calculation limits. That's not that difficult. I wrote a VBA routine to calculate 100!. I just assign a variable to each digit, and do the multiplying digit-by-digit. Something of a pain, but it does easily get around the 15 or so digit limits of Excel. There are other software packages you can get to handle large number arithmetic, but I haven't had to resort to those yet.