@inproceedings{08ccb3e60cbd498db152b04b8f97e21d,

title = "On the complexity of computing prime tables",

abstract = "Many large arithmetic computations rely on tables of all primes less than n. For example, the fastest algorithms for computing n! takes time O(M(n log n) + P(n)), where M(n) is the time to multiply two n-bit numbers, and P(n) is the time to compute a prime table up to n. The fastest algorithm to compute (nn/2) also uses a prime table. We show that it takes time O(M(n) + P(n)). In various models, the best bound on P(n) is greater than M(n log n), given advances in the complexity of multiplication [8,13]. In this paper, we give two algorithms to computing prime tables and analyze their complexity on a multitape Turing machine, one of the standard models for analyzing such algorithms. These two algorithms run in time O(M(n log n)) and O(n log2 n/ log log n), respectively. We achieve our results by speeding up Atkin{\textquoteright}s sieve. Given that the current best bound on M(n) is n log n2 O(log ∗ n), the second algorithm is faster and improves on the previous best algorithm by a factor of log2 log n. Our fast prime-table algorithms speed up both the computation of n! and (n/n2). Finally, we show that computing the factorial takes Ω(M(n log4/7−εn)) for any constant ε > 0 assuming only multiplication is allowed.",

keywords = "Factorial, Lower bound, Multiplication, Prime tables",

author = "Mart{\'i}n Farach-Colton and Meng-Tsung Tsai",

year = "2015",

month = jan,

day = "1",

doi = "10.1007/978-3-662-48971-0_57",

language = "English",

isbn = "9783662489703",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer Verlag",

pages = "677--688",

editor = "Khaled Elbassioni and Kazuhisa Makino",

booktitle = "Algorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings",

address = "Germany",

note = "null ; Conference date: 09-12-2015 Through 11-12-2015",

}