The Pseudosquares Prime Sieve

The Pseudosquares Prime Sieve

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by Jonathan P. Sorenson

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Datum	Value	Source
TITLE	The Pseudosquares Prime Sieve	SVM HeaderParse 0.2
AUTHOR NAME	Jonathan P. Sorenson	SVM HeaderParse 0.2
AUTHOR AFFIL	Computer Science and Software Engineering,; Butler University	SVM HeaderParse 0.2
AUTHOR ADDR	Indianapolis, IN 46208 USA	SVM HeaderParse 0.2
ABSTRACT	Abstract. 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)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) 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 nlog log n. Thus, log2 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 ≤ n 1/(4√e−ɛ) 1.132, implying a running time of roughly n using roughly n 0.132 space. Existing prime sieves are generally faster but take much more space, greatly limiting their range (O(n / log log n)operationswithn 1/3+ɛ space, or O(n) operationswithn 1/4 conjectured space). Our algorithm found all 13284 primes in the interval [10 33,10 33 +10 6] in about 4 minutes on a1.3GHzPentiumIV. We also present an algorithm to find all pseudosquares Lp up to n in sublinear time using very little space. Our innovation here is a new, space-efficient implementation of the wheel datastructure. 1	SVM HeaderParse 0.2
CITATIONS	28 found	ParsCit 1.0