One way to sort the shaft from the treasure is to take out the bad stuff.........everything that is not a prime.....like a sculptor widdles away at a stone......to make whatever.......


So...............one way of looking at big numbers.........................and any series of numbers beyond single digits...........for primes...................to eliminate them.........is to eliminate anything even and anything that ends in a 5.................eliminating anything even.....halfs the entire number line...................

Take out anything that ends in 5...........and for any block of ten consecutive numbers.....................e.g..................220 - 230...............u have only 4 possible numbers which could be prime..............so.......

220 221 222 223 224 225 226 227 228 229 230.....................anything that ends in a zero are divisible by two..................even..........would be eliminated, along with 225........b/c it ends in 5.........

leaving..............221 223 227 229..........in fact, in any series of 10 consecutive numbers..........excluding single digits....................the only possible 4 numbers with a chance for being prime........would end in 1, 3, 7, or 9......................take 30 - 40...................30, 32, 34, 35, 36, 38, and 40 could not be prime.............only numbers that end in 1, 3, 7 or a 9..........above single digits have a chance at being prime........




The ancient Greeks proved (ca 300 BC) that there were infinitely many primes and that they were irregularly spaced (there can be arbitrarily large gaps between successive primes).  On the other hand, in the nineteenth century it was shown that the number of primes less than or equal to n approaches n/(log n) (as n gets very large); so a rough estimate for the nth prime is n log n (see the document "How many primes are there?")
The 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.

The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic.  It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length...  Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. (Carl Friedrich Gauss, Disquisitiones Arithmeticae, 1801)