Prime Numbers And Primality Tests

Prime Numbers And Primality TestsTesting PrimalityEuclids ProofInteger Factorization

What is Prime Number ?Prime Number:- The number which is divisible by 1 or itself only.

Example :- 2,3,5,7,11,13.

Primality Test:-A Primality Test is an algorithm to determining whether an input number is prime or not .Trial Division:- It is a simplest primality test .Given an input numbern, check whether any integermfrom 2 to n 1 evenlydividesn(the division leaves noremainder). Ifnis divisible by anymthenniscomposite, otherwise it isprime.Naive method:-bool isPrime(unsigned long n) { if (n 1; } else if (n % 2 == 0 || n % 3 == 0) { return false; }******else { for (unsigned short i = 5; i * i info=left13 break14 return(fact)

ANOTHER APPROACH:-TRIALL DIVISION METHOD:-Divide n with all primes up to n starting from 2 and collect all divisors. A very simple algorithm. Takes time exp( log n) = L(1, ).Notation: Denote exp(c(log n)(loglog n)1-) as L(, c).

APPLICATION OF I.F :-CRYPTOGRAPHY - ATM-COMPUTER SCIENCE -AND ELECTRONIC

REFRENCES:-Richard Crandall and Carl Pomerance (2001). Prime Numbers: A Computational PerspectiveDonald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms , Third :Factoring into Primes.www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdfKleinjung, et al (2010-02-18). "Factorization of a 768-bit RSA modulus". International Association for Cryptologic Research. Retrieved 2010-08-09. Euclid's Elements, Book IX, Prop. 20 (Euclid's proof, on David Joyce's website at Clark University)Weisstein, Eric W., "Euclid's Theorem", MathWorld.

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