A new prime for which the least primitive root and the least primitive root are not equal
Author:
A. Paszkiewicz
Journal:
Math. Comp. 78 (2009), 11931195
MSC (2000):
Primary 11Y16; Secondary 11A07, 11M26
Published electronically:
October 31, 2008
MathSciNet review:
2476579
Fulltext PDF
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Abstract: With the aid of a computer network we have performed a search for primes and revealed a new prime for which its least primitive root and its least primitive root are not equal.
 1.
N. Beeger, On a new case of the congruence . Messenger of Mathematics, 51 (1922), pp. 149150.
 2.
Richard
Crandall, Karl
Dilcher, and Carl
Pomerance, A search for Wieferich and Wilson
primes, Math. Comp. 66
(1997), no. 217, 433–449. MR 1372002
(97c:11004), http://dx.doi.org/10.1090/S0025571897007916
 3.
Wilfrid
Keller and Jörg
Richstein, Solutions of the congruence
𝑎^{𝑝1}≡1\pmod{𝑝^{𝑟}}, Math. Comp. 74 (2005), no. 250, 927–936 (electronic). MR 2114655
(2005i:11004), http://dx.doi.org/10.1090/S0025571804016667
 4.
Joshua
Knauer and Jörg
Richstein, The continuing search for Wieferich
primes, Math. Comp. 74
(2005), no. 251, 1559–1563
(electronic). MR
2137018 (2006a:11006), http://dx.doi.org/10.1090/S0025571805017230
 5.
E.
L. Litver and G.
E. Judina, Primitive roots for the first million primes and their
powers, Mathematical analysis and its applications, Vol. III
(Russian), Izdat. Rostov. Univ., RostovonDon, 1971,
pp. 106–109 (Russian). MR 0340159
(49 #4915)
 6.
W. Meissner, Über die Teilbarkeit von durch das Quadrat der Primzahl , Sitzungsberichte Preuss. Akad. Wiss. (1913), pp. 663667.
 7.
Peter
L. Montgomery, New solutions of
𝑎^{𝑝1}≡1\pmod{𝑝²}, Math. Comp. 61 (1993), no. 203, 361–363. MR 1182246
(94d:11003), http://dx.doi.org/10.1090/S00255718199311822465
 8.
A.
Paszkiewicz and A.
Schinzel, Numerical calculation of the density
of prime numbers with a given least primitive root, Math. Comp. 71 (2002), no. 240, 1781–1797 (electronic). MR 1933055
(2003g:11109), http://dx.doi.org/10.1090/S0025571801013825
 1.
 N. Beeger, On a new case of the congruence . Messenger of Mathematics, 51 (1922), pp. 149150.
 2.
 R. Crandall, K. Dilcher, C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp., 1997, 66, pp. 433449. MR 1372002 (97c:11004)
 3.
 W. Keller, J. Richstein, Solutions of the congruence , Math. Comp., 2005, 74, pp. 927936. MR 2114655 (2005i:11004)
 4.
 J. Knauer, J. Richstein, The continuing search for Wieferich primes, Math. Comp., 2005, 74, pp. 15591563. 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. 106109. MR 0340159 (49:4915)
 6.
 W. Meissner, Über die Teilbarkeit von durch das Quadrat der Primzahl , Sitzungsberichte Preuss. Akad. Wiss. (1913), pp. 663667.
 7.
 P. L. Montgomery, New solutions of , Math. Comp. 61 (1993), 361363. 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. 17811797. MR 1933055 (2003g:11109)
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Additional Information
A. Paszkiewicz
Affiliation:
Warsaw University of Technology, Institute of Telecommunications, ul.\ Nowowiejska 15/19, 00665 Warsaw, Poland
Email:
anpa@tele.pw.edu.pl
DOI:
http://dx.doi.org/10.1090/S0025571808020905
PII:
S 00255718(08)020905
Keywords:
Prime generators,
primitive roots
Received by editor(s):
November 15, 2004
Received by editor(s) in revised form:
July 27, 2007
Published electronically:
October 31, 2008
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
