OFFSET
1,1
COMMENTS
For r a primitive root of a prime p, r + qp is a primitive root of p: but r + qp is also a primitive root of p^2, except for q in some unique residue class modulo p. In the exceptional case, r + qp has order p-1 modulo p^2 (Burton, section 8.3).
No other terms below 10^12 (Paszkiewicz, 2009).
Each term p is a Wieferich prime to base A046145(p). For example, a(2) and a(3) are base-5 Wieferich (see A123692). - Jeppe Stig Nielsen, Mar 06 2020
REFERENCES
David Burton, Elementary Number Theory, Allyn and Bacon, Boston, 1976, first edition (cf. Section 8.3).
LINKS
Joerg Arndt, Matters Computational (The Fxtbook), section 39.7.2, p.780.
Stephen Glasby, Three questions about the density of certain primes, Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Apr 22, 2001.
Bryce Kerr, Kevin McGown, Tim Trudgian, The least primitive root modulo p^2, arXiv:1908.11497 [math.NT], 2019.
A. Paszkiewicz, A new prime for which the least primitive root (mod p) and the least primitive root (mod p^2) are not equal, Math. Comp. 78 (2009), 1193-1195.
FORMULA
MATHEMATICA
Select[Prime@Range[7!], ! PrimitiveRoot[#] == PrimitiveRoot[#^2] &] (* Arkadiusz Wesolowski, Sep 06 2012 *)
CROSSREFS
KEYWORD
hard,nonn,bref,more
AUTHOR
Bernard Leak (bernard(AT)brenda-arkle.demon.co.uk), Aug 24 2000
EXTENSIONS
a(3) from Stephen Glasby (Stephen.Glasby(AT)cwu.EDU), Apr 22 2001
Edited by Max Alekseyev, Nov 10 2011
STATUS
approved