A055578 "Non-generous primes": primes p whose least positive primitive root is not a primitive root of p^2.

2, 40487, 6692367337

Offset: 1

Author

Bernard Leak (bernard(AT)brenda-arkle.demon.co.uk), Aug 24 2000

Keywords

• hard
• nonn
• bref
• more

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.

Crossrefs

Cf. A060503, A060504.

Programs

Mathematica

Select[Prime@Range[7!], ! PrimitiveRoot[#] == PrimitiveRoot[#^2] &] (* Arkadiusz Wesolowski, Sep 06 2012 *)

Formula

Prime A000040(n) is in this sequence iff A001918(n)^(A000040(n)-1) == 1 (mod A000040(n)^2).

Prime A000040(n) is in this sequence iff A001918(n) differs from A127807(n).

Extensions

a(3) from Stephen Glasby (Stephen.Glasby(AT)cwu.EDU), Apr 22 2001

Edited by Max Alekseyev, Nov 10 2011