Mathematics of Computation

A new prime

for which the least primitive root $({\rm mod} p)$ and the least primitive root $({\rm mod} p^2)$ are not equal

Author(s): A. Paszkiewicz.
Journal: Math. Comp. 78 (2009), 1193-1195.
MSC (2000): Primary 11Y16; Secondary 11A07, 11M26
Posted: October 31, 2008
Retrieve article in: PDF

Abstract: With the aid of a computer network we have performed a search for primes $p<10^{12}$ and revealed a new prime

for which its least primitive root $({\rm mod} p)$ and its least primitive root $({\rm mod} p^2)$ are not equal.

1.: N. Beeger, On a new case of the congruence $2^{p-1}\equiv 1 ({\rm mod} p^2)$ . Messenger of Mathematics, 51 (1922), pp. 149-150.
2.: R. Crandall, K. Dilcher, C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp., 1997, 66, pp. 433-449. MR 1372002 (97c:11004)
3.: W. Keller, J. Richstein, Solutions of the congruence $a^{p-1}\equiv 1 ({\rm mod} p^r)$ , Math. Comp., 2005, 74, pp. 927-936. MR 2114655 (2005i:11004)
4.: J. Knauer, J. Richstein, The continuing search for Wieferich primes, Math. Comp., 2005, 74, pp. 1559-1563. MR 2137018 (2006a:11006)
5.: E. L. Litver, G. E. Yudina, Primitive roots for the first million primes and their powers (Russian), in: Matematiceskij analiz i ego prilozenija, III Rostov 1971, pp. 106-109. MR 0340159 (49:4915)
6.: W. Meissner, Über die Teilbarkeit von durch das Quadrat der Primzahl , Sitzungsberichte Preuss. Akad. Wiss. (1913), pp. 663-667.
7.: P. L. Montgomery, New solutions of $a^{p-1}\equiv 1 ({\rm mod} p^2)$ , Math. Comp. 61 (1993), 361-363. MR 1182246 (94d:11003)
8.: A. Paszkiewicz, A. Schinzel, Numerical calculation of the density of prime numbers with a given least primitive root, Math. Comp., 2002, 71, pp. 1781-1797. MR 1933055 (2003g:11109)

Retrieve articles in Mathematics of Computation with MSC (2000): 11Y16, 11A07, 11M26

A. Paszkiewicz
Affiliation: Warsaw University of Technology, Institute of Telecommunications, ul. Nowowiejska 15/19, 00-665 Warsaw, Poland
Email: anpa@tele.pw.edu.pl

DOI: 10.1090/S0025-5718-08-02090-5
PII: S 0025-5718(08)02090-5
Keywords: Prime generators, primitive roots
Received by editor(s): November 15, 2004
Received by editor(s) in revised form: July 27, 2007
Posted: October 31, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS