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SYLOW p-PSEUDOPRIMES TO SEVERAL BASES FOR SEVERAL PRIMES p

Part of: Computational number theory Elementary number theory

Published online by Cambridge University Press:  05 October 2009

ZHENXIANG ZHANG  and
RUIRUI XIE
ZHENXIANG ZHANG*
Affiliation:
Department of Mathematics, Anhui Normal University, 241000 Wuhu, Anhui, PR China (email: zhangzhx@mail.wh.ah.cn, ahnu_zzx@sina.com)
RUIRUI XIE
Affiliation:
Department of Mathematics, Anhui Normal University, 241000 Wuhu, Anhui, PR China (email: XieRR19860@163.com)
*
✉For correspondence; e-mail: zhangzhx@mail.wh.ah.cn, ahnu_zzx@sina.com
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Abstract

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Browkin [‘Some new kinds of pseudoprimes’, Math. Comp. 73 (2004), 1031–1037] gave examples of strong pseudoprimes to many bases which are not Sylow p-pseudoprimes to two bases only, where p=2 or 3. In contrast to Browkin’s examples, Zhang [‘Notes on some new kinds of pseudoprimes’, Math. Comp. 75 (2006), 451–460] gave facts and examples which are unfavorable for Browkin’s observation on detecting compositeness of odd composite numbers. In particular, Zhang gave a Sylowp-pseudoprime (with 27 decimal digits) to the first 6 prime bases for all the first 6 primes p, and conjectured that for any k≥1, there would exist Sylow p-pseudoprimes to the first k prime bases for all the first k primes p. In this paper we tabulate 27 Sylow p-pseudoprimes less than 1036 to the first 7 prime bases for all the first 7 primes p (two of which are Sylow p-pseudoprimes to the first 7 prime bases for all the first 8 primes p). We describe the procedure for finding these numbers. The main tools used in our method are the cubic residue characters and the Chinese remainder theorem.

Keywords

strong pseudoprimesMiller testsSylow p-pseudoprimescubic residue charactersChinese remainder theorem

MSC classification

Secondary: 11A15: Power residues, reciprocity 11Y11: Primality
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 1 , February 2010 , pp. 165 - 176
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

Research supported by the NSF of China Grant 10071001.

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