Lazy Wheel Sieves and Spirals of Primes

Trading Time for Space in Prime Number Sieves

A prime number sieve is an algorithm that nds the primes up to a bound n. We present four new prime number sieves. Each of these sieves gives new space complexity bounds for certain ranges of running times. In particular, we give a linear time sieve that uses only O(p n=(log log n) 2) bits of space, an O l (n= log log n) time sieve that uses O(n=((log n) l log log n)) bits of space, where l > 1...

Ramanujan and Labos Primes, Their Generalizations, and Classifications of Primes

We study the parallel properties of the Ramanujan primes and a symmetric counterpart, the Labos primes. Further, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems.

The Pseudosquares Prime Sieve

We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp > n/(π(p)(logn)); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p) log n) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Ri...

Two Fast Parallel Prime Number Sieves

On the Small Sieve . I . Sifting by Primes

The main object of the paper is to prove that if P is a set of primes with sum of reciprocals _cx, where c is a positive constant depending only on K. A lower estimate is given for c and a similar result is achieved in the case when the condition of primality is substituted by the weaker condition that any m elem...