Wheel factorization

Abstract: Prime numbers play a pivotal role in current encryption algorithms and given the rise of cloud computing, the need for larger primes has never been so high This increase in available computation power can be used to either try to break the encryption or to strength it by finding larger prime numbers With this in mind, this paper provides an analysis of different sieve implementations that can be used to generate primes to near 264 It starts by analyzing cache friendly sequential sieves with wheel factorization, then expands to multi-core architectures and ends with a cache friendly segmented hybrid implementation of a distributed prime sieve, designed to efficiently use all the available computation resources of heterogeneous computer clusters with variable workload and to scale very well in both the shared and distributed memory versions

Abstract: In this document, prime numbers are related as functions over time, mimicking the Sieve of Eratosthenes. For this purpose, the mathematical representation is a uni-dimentional time line depicting the number line for positive natural numbers N, where each number n represents a time t. In the same way as the Eratosthenes' sieve, which iteratively mark as composite the multiples of each prime, starting at each prime. This dynamical prime number function P(s) zero-cross all composite numbers departing from primes, following a linear progression over time.

Abstract: The Sieve of Eratosthenes has been recently extended by excluding the multiples of 2, 3, and 5 from the initial set, and finding the additive rules that give the positions of the multiples of the remaining primes. We generalize these results. For a given k we let the initial set S k consists of natural numbers relatively prime to the first k primes, and find the rules governing the positions of the multiples of the remaining elements.