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

2, 40487, 6692367337

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

Table of n, a(n) for n=1..3.

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

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).

Mathematica

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

Crossrefs

Cf. A060503, A060504.

Sequence in context: A291881 A257968 A303738 * A232733 A106025 A157959

Adjacent sequences: A055575 A055576 A055577 * A055579 A055580 A055581

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