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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The Lucas-Pratt primality tree


Author: Jonathan Bayless
Journal: Math. Comp. 77 (2008), 495-502
MSC (2000): Primary 11Y16; Secondary 11N37
Published electronically: May 14, 2007
MathSciNet review: 2353963
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Abstract: In 1876, E. Lucas showed that a quick proof of primality for a prime $ p$ could be attained through the prime factorization of $ p-1$ and a primitive root for $ p$. V. Pratt's proof that PRIMES is in NP, done via Lucas's theorem, showed that a certificate of primality for a prime $ p$ could be obtained in $ O(\log^2 p)$ modular multiplications with integers at most $ p$. We show that for all constants $ C \in \mathbb{R}$, the number of modular multiplications necessary to obtain this certificate is greater than $ C \log p$ for a set of primes $ p$ with relative asymptotic density 1.


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Additional Information

Jonathan Bayless
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hamphshire 03755-3551
Email: jonathan.bayless@dartmouth.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-07-02002-9
PII: S 0025-5718(07)02002-9
Received by editor(s): June 26, 2006
Received by editor(s) in revised form: November 14, 2006
Published electronically: May 14, 2007
Additional Notes: The author was supported by a Dartmouth Graduate Fellowship
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.