Prime Numbers History, Facts and Examples - Prime Numbers An Introduction Prime number is the number which is greater than 1 and cannot be divided

# Prime Numbers History, Facts and Examples - Prime...

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Prime Numbers: An Introduction Prime number is the number, which is greater than 1 and cannot be divided by any number excluding itself and one. A prime number is a positive integer that has just two positive integer factors, including 1 and itself. Such as, if the factors of 28 are listed, there are 6 factors that are 1, 2, 4, 7, 14, and 28. Similarly, if the factors of 29 are listed, there are only two factors that are 1 and 29. Therefore, it can be inferred that 29 is a prime number, but 28 is not. Examples of prime numbers The first few prime numbers are as follows: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc. Identifying the primes The ancient Sieve of Eratosthenes is a simple way to work out all prime numbers up to a given limit by preparing a list of all integers and repetitively striking out multiples of already found primes. There is also a modern Sieve of Atkin, which is more complex when compared to that of Eratosthenes. A method to determine whether a number is prime or not, is to divide it by all primes less than or equal to the square root of that number. If the results of any of the divisions are an integer, the original number is not a prime and if not, it is a prime. One need not actually calculate the square root; once one sees that the quotient is less than the divisor, one can stop. This is called as the trial division, which is the simplest primality test but it is impractical for testing large integers because the number of possible factors grows exponentially as the number of digits in the number to be tested increases. Primality tests: A primality test algorithm is an algorithm that is used to test a number for primality, that is, whether the number is a prime number or not. AKS primality test The AKS primality test is based upon the equivalence (x - a) n = (x n - a) (mod n) for a coprime to n, which is true if and only if n is prime. This is a generalization of Fermat's little theorem extended to polynomials and can easily be proven using the binomial theorem together with the fact that: for all 0 < k < n if n is prime. While this equivalence constitutes a primality test in itself, verifying it takes exponential time. Therefore AKS makes use of a related equivalence (x - a) n = (x n - a) (mod n, x r - 1), which can be checked in polynomial time. Fermat primality test Fermat's little theorem asserts that if p is prime and 1 a < p, then a p -1 ≡ 1 (mod p) In order to test whether p is a prime number or not, one can pick random a's in the interval and check if there is an equality. Solovay-Strassen primality test
For a prime number p and any integer a, A (p -1)/2 ≡ (a/p) (mod p) Where (a/p) is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to (a/n); where n can be any odd integer. The Jacobi symbol can be computed in time O((log n)²) using Jacobi's generalization of law of quadratic reciprocity. It can be observed whether or not the congruence A (n -1)/2 ≡ (a/n) (mod n) holds for various values of a. This congruence is true for all a's if n is a prime number. (Solovay, Robert M. and Volker Strassen, 1977)
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