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/((p)(log n)2 ); here (x) is the prime counting function. Our algorithm requires only O((p)n) arithmetic operations and O((p) log n) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Riemann Hypothesis, we have p 2(log n)2 , but it is conjectured that p 1 log 2 log n log log n. Thus, the conjectured complexity of our prime sieve is O(n log n) arithmetic operations in O((log n)2 ) space. The primes generated by our algorithm are proven prime unconditionally. The best current unconditional bound known is p n1/(4 e- )

• Type: Conference
• Year: 2006
• Where: ANTS
• Authors: Jonathan Sorenson
• Keywords: Algorithms, ANTS 2006, Arithmetic Operations, Prime Sieve, Pseudosquares Prime Sieve