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


Finding strong pseudoprimes to several bases. II

Authors: Zhenxiang Zhang and Min Tang
Journal: Math. Comp. 72 (2003), 2085-2097
MSC (2000): Primary 11Y11, 11A15, 11A51
Published electronically: May 30, 2003
MathSciNet review: 1986825
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Abstract: Define $\psi_m$ to be the smallest strong pseudoprime to all the first $m$ prime bases. If we know the exact value of $\psi_m$, we will have, for integers $n<\psi_m$, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the $\psi_m$ are known for $1 \leq m \leq 8$. Upper bounds for $\psi_9,\psi_{10} \text{ and } \psi_{11}$ were first given by Jaeschke, and those for $\psi_{10} \text{ and } \psi_{11}$ were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863-872).

In this paper, we first follow the first author's previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp's) $n<10^{24}$ to the first five or six prime bases, which have the form $n=p\,q$ with $p, q$ odd primes and $q-1=k(p-1), k=4/3,\,5/2,\,3/2,\,6$; then we tabulate all Carmichael numbers $<10^{20}$, to the first six prime bases up to 13, which have the form $n=q_1q_2q_3$ with each prime factor $q_i\equiv 3\mod 4$. There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp's to base 17; 5 numbers are spsp's to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for $\psi_{9}, \psi_{10}$ and $\psi_{11}$ are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number:

\begin{displaymath}\begin{split} \psi_{9}\leq \psi_{10}\leq \psi_{11}\leq Q_{11}... ...(19 digits)}\\ &= 149491\cdot 747451\cdot 34233211. \end{split}\end{displaymath}

We conjecture that

\begin{displaymath}\psi_{9}= \psi_{10}= \psi_{11}=3825\;12305\;65464\;13051,\end{displaymath}

and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor $q_3$ and propose necessary conditions on $n$ to be a strong pseudoprime to the first $5$ prime bases. Comparisons of effectiveness with Arnault's, Bleichenbacher's, Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.

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Additional Information

Zhenxiang Zhang
Affiliation: Department of Mathematics, Anhui Normal University, 241000 Wuhu, Anhui, Peoples Republic of China

Min Tang
Affiliation: Department of Mathematics, Anhui Normal University, 241000 Wuhu, Anhui, Peoples Republic of China

PII: S 0025-5718(03)01545-X
Keywords: Strong pseudoprimes, Carmichael numbers, Rabin-Miller test, biquadratic residue characters, cubic residue characters, Chinese remainder theorem
Received by editor(s): July 9, 2001
Published electronically: May 30, 2003
Additional Notes: Supported by the NSF of China Grant 10071001, the SF of Anhui Province Grant 01046103, and the SF of the Education Department of Anhui Province Grant 2002KJ131.
Article copyright: © Copyright 2003 American Mathematical Society

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