Notes on some new kinds of pseudoprimes

Author:
Zhenxiang Zhang

Journal:
Math. Comp. **75** (2006), 451-460

MSC (2000):
Primary 11A15; Secondary 11A51, 11Y11

DOI:
https://doi.org/10.1090/S0025-5718-05-01775-8

Published electronically:
September 15, 2005

MathSciNet review:
2176408

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Abstract | References | Similar Articles | Additional Information

Abstract: J. Browkin defined in his recent paper (Math. Comp. **73** (2004), pp. 1031-1037) some new kinds of pseudoprimes, called Sylow -pseudoprimes and elementary Abelian -pseudoprimes. He gave examples of strong pseudoprimes to many bases which are not Sylow -pseudoprime to two bases only, where or .

In this paper, in contrast to Browkin's examples, we give facts and examples which are unfavorable for Browkin's observation to detect compositeness of odd composite numbers. In Section 2, we tabulate and compare counts of numbers in several sets of pseudoprimes and find that most strong pseudoprimes are also Sylow -pseudoprimes to the same bases. In Section 3, we give examples of Sylow -pseudoprimes to the first several prime bases for the first several primes . We especially give an example of a strong pseudoprime to the first six prime bases, which is a Sylow -pseudoprime to the same bases for all . In Section 4, we define to be a -*fold* *Carmichael Sylow pseudoprime*, if it is a Sylow -pseudoprime to all bases prime to for all the first smallest odd prime factors of . We find and tabulate all three -fold Carmichael Sylow pseudoprimes . In Section 5, we define a positive odd composite to be a *Sylow uniform pseudoprime to bases* , or a Syl-upsp for short, if it is a Syl-psp for all the first small prime factors of , where is the number of distinct prime factors of . We find and tabulate all the 17 Syl-upsp's and some Syl-upsp 's . Comparisons of effectiveness of Browkin's observation with Miller tests to detect compositeness of odd composite numbers are given in Section 6.

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Additional Information

**Zhenxiang Zhang**

Affiliation:
Department of Mathematics, Anhui Normal University, 241000 Wuhu, Anhui, People’s Republic of China

Email:
zhangzhx@mail.ahwhptt.net.cn, ahnu_zzx@sina.com

DOI:
https://doi.org/10.1090/S0025-5718-05-01775-8

Keywords:
Strong pseudoprimes,
Miller tests,
Sylow $p$-pseudoprimes,
elementary Abelian $p$-pseudoprimes,
$k$-fold Carmichael Sylow pseudoprimes,
Sylow uniform pseudoprimes

Received by editor(s):
September 18, 2004

Published electronically:
September 15, 2005

Additional Notes:
This work was supported by the NSF of China Grant 10071001, and the SF of the Education Department of Anhui Province Grant 2002KJ131

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.