044-pus
the prime number are oth odd.
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Answer:
How do we go about finding primes? And once we have found them, how do we prove they are truly prime? The answer depends on the size of the primes and how sure we need to be of their primality. In these pages we present the appropriate answers in several sections. Let us preview these chapters one at a time.
Chapter Two: The quick tests for small numbers and probable primes
For very small primes we can use the Sieve of Eratosthenes or trial division. These methods are sure, and are the best methods for small numbers, but become far too time consuming before our numbers reach thirty digits.
If we are going to use our primes for "industrial" uses (e.g., for RSA encryption) we often do not need to prove they are prime. It may be enough to know that the probability they are composite is less than 0.000000000000000000000001%. In this case we can use (strong) probable primality tests.
These probable primality tests can be combined to create a very quick algorithm for proving primality for integers less than 340,000,000,000,000.
Chapter Three: The classical tests
A quick look at the list of largest known primes shows numbers with hundreds of thousands (even millions) of digits--and these are all proven primes (not probable primes)! So how can we know they are prime? Look at a portion of this list and decide what the numbers all have in common.
111 189*2^34233-1 10308 Z 89
112 15*2^34224+1 10304 D 93
113 (5452545+10^5153)*10^5147+1 10301 D 90 Palindrome
114 23801#+1 10273 C 93 primorial plus one
115 63*2^34074+1 10260 Y 95
116 213819*2^33869+1 10201 Y 93
They are all trivial to factor if we either add, or subtract, one! This is no accident.
It is possible to turn the probable-primality tests of chapter two for an integer n into primality proofs, if we know enough factors of either n+1 and/or n-1. These proofs are called the classical tests and we survey them in our third chapter.
These tests have been used for over 99.99% of the largest known primes. They include special cases such as the Lucas-Lehmer test for Mersenne primes and Pepin's Test for Fermat primes.