turing programming language list first n primes

And we hope that careful reading will lead you to be a better programmer in addition to learning something about prime numbers. We describe algorithms that are well known and implement them carefully.

Our purpose is pedagogical, so we are primarily interested in the clarity of the code. We first describe the algorithms in the body of the essay, then describe actual implementations in five languages - C, Haskell, Java, Python and Scheme - in a series of appendices. We describe in detail five relevant functions: one that makes a list of the prime numbers less than a given number using the Sieve of Eratosthenes two that determine whether a given number is prime or composite, one using trial division and the other using an algorithm developed by Gary Miller and Michael Rabin and two that find the unique factorization of a given composite number, one using trial division and the other using John Pollard’s rho algorithm. In this essay we will examine three problems related to prime numbers: enumerating the prime numbers, determining if a given number is prime or composite, and factoring a composite number into its prime factors.

Euclid, Elements, Book IX, Proposition 20, circa 300 B.C. In either case the assumption that the number of primes is finite is contradicted, thus proving the infinitude of primes. Thus either n is prime, or n is composite but has two or more prime factors not on the list p 1, p 2, …, p k of prime numbers.

Certainly n is not evenly divisible by any of the primes p 1, p 2, …, p k, because division by any them leaves a remainder of 1. The sequence of prime numbers begins 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, … and continues to infinity, which Euclid, the famous teacher of geometry, proved about twenty-three centuries ago:Īssume for the moment that the number of primes is finite, and make a list of them: p 1, p 2, …, p k. Prime numbers have fascinated mathematicians since the days of the ancient Greek mathematicians, and remain an object of study today. Prime numbers are those integers greater than one that are divisible only by themselves and one an integer greater than one that is not prime is composite.

Turing programming language list first n primes