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Mathematics of Computation

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Counting composites with two strong liars

Authors: Eric Bach and Andrew Shallue
Journal: Math. Comp. 84 (2015), 3069-3089
MSC (2010): Primary 11Y11
Published electronically: April 1, 2015
MathSciNet review: 3378863
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Abstract: The strong probable primality test is an important practical tool for discovering prime numbers. Its effectiveness derives from the following fact: for any odd composite number $ n$, if a base $ a$ is chosen at random, the algorithm is unlikely to claim that $ n$ is prime. If this does happen we call $ a$ a liar. In 1986, Erdős and Pomerance computed the normal and average number of liars, over all $ n \leq x$. We continue this theme and use a variety of techniques to count $ n \leq x$ with exactly two strong liars, those being the $ n$ for which the strong test is maximally effective. We evaluate this count asymptotically and give an improved algorithm to determine it exactly. We also provide asymptotic counts for the restricted case in which $ n$ has two prime factors, and for the $ n$ with exactly two Euler liars.

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Additional Information

Eric Bach
Affiliation: University of Wisconsin–Madison, 1210 W. Dayton St., Madison, Wisconsin 53706

Andrew Shallue
Affiliation: Illinois Wesleyan University, 1312 Park St., Bloomington, Illinois 61701

Received by editor(s): August 4, 2013
Received by editor(s) in revised form: February 16, 2014
Published electronically: April 1, 2015
Additional Notes: This research supported by NSF: CCF-0635355 and ARO: W911NF9010439
Article copyright: © Copyright 2015 American Mathematical Society

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