Sieves..............take the rocks from the sand........my grandfather made me do that.......so he could make cement........


he Sieve of Eratosthenes is still the most efficient way of finding all very small primes (e.g., those less than 1,000,000).  However, most of the largest primes are found using special cases of Lagrange's Theorem from group theory.  See the separate documents on proving primality for more information.
In 1984 Samuel Yates defined a titanic prime to be any prime with at least 1,000 digits [Yates84, Yates85].  When he introduced this term there were only 110 such primes known; now there are over 1000 times that many!  And as computers and cryptology continually give new emphasis to search for ever larger primes, this number will continue to grow.   Before long we expect to see the first twenty-five million digit prime.
If you want to understand a building, how it will react to weather or fire, you first need to know what it is made of. The same is true for the integers--most of their properties can be traced back to what they are made of: their prime factors. For example, in Euclid's Geometry (over 2,000 years ago), Euclid studied even perfect numbers and traced them back to what we now call Mersenne primes.