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

DOI:
https://doi.org/10.1090/S0025-5718-03-01545-X

Published electronically:
May 30, 2003

MathSciNet review:
1986825

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Abstract: Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Upper bounds for were first given by Jaeschke, and those for 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) to the first five or six prime bases, which have the form with odd primes and ; then we tabulate all Carmichael numbers , to the first six prime bases up to 13, which have the form with each prime factor . 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 and are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number:

We conjecture that

and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor and propose necessary conditions on to be a strong pseudoprime to the first 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

Email:
zhangzhx@mail.ahwhptt.net.cn

**Min Tang**

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

DOI:
https://doi.org/10.1090/S0025-5718-03-01545-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