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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Infinite sets of primes with fast primality tests and quick generation of large primes
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by János Pintz, William L. Steiger and Endre Szemerédi PDF
Math. Comp. 53 (1989), 399-406 Request permission


Infinite sets P and Q of primes are described, $P \subset Q$. For any natural number n it can be decided if $n \in P$ in (deterministic) time $O({(\log n)^9})$. This answers affirmatively the question of whether there exists an infinite set of primes whose membership can be tested in polynomial time, and is a main result of the paper. Also, for every $n \in Q$, we show how to randomly produce a proof of the primality of n. The expected time is that needed for $1\frac {1}{2}$ exponentiations $\bmod n$. We also show how to randomly generate k-digit integers which will be in Q with probability proportional to ${k^{ - 1}}$. Combined with the fast verification of $n \in Q$ just mentioned, this gives an $O({k^4})$ expected time algorithm to generate and certify primes in a given range and is probably the fastest method to generate large certified primes known to belong to an infinite subset. Finally, it is important that P and Q are relatively dense (at least $c{n^{2/3}}/\log n$ elements less than n). Elements of Q in a given range may be generated quickly, but it would be costly for an adversary to search Q in this range, a property that could be useful in cryptography.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Math. Comp. 53 (1989), 399-406
  • MSC: Primary 11Y11; Secondary 11Y16
  • DOI:
  • MathSciNet review: 968154