OFFSET
1,1
COMMENTS
Primes p with a primitive root g such that g^2 = g + 1 (mod p).
Not the same as primes with a Fibonacci number as primitive root; cf. A083701. - Jonathan Sondow, Feb 17 2013
For all except the initial term 5, these are numbers such that the Pisano period equals 1 less than the Pisano number, i.e., where A001175(n) = n-1. - Matthew Goers, Sep 20 2013
As shown in the paper by Brison, these are also the primes p such that there is a Fibonacci-type sequence (mod p) that begins with (1,b) and encounters all numbers less than p in the first p-1 iterations (for some b). - T. D. Noe, Feb 26 2014
Shanks (1972) conjectured that the relative asymptotic density of this sequence in the sequence of primes is 27*c/38 = 0.2657054465..., where c is Artin's constant (A005596). The conjecture was proved on the assumption of a generalized Riemann hypothesis by Lenstra (1977) and Sander (1990). - Amiram Eldar, Jan 22 2022
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Noe)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly, Vol. 58, No. 5 (2020), pp. 25-29.
Owen J. Brison, Complete Fibonacci sequences in finite fields, Fibonacci Quarterly, Vol. 30, No. 4 (1992), pp. 295-304.
Alexandru Gica, Quadratic Residues in Fibonacci Sequences, Fibonacci Quart., Vol. 46/47, No. 1 (2008/2009), pp. 68-72. See Theorem 5.1.
Liang-Chung Hsia, Hua-Chieh Li, and Wei-Liang Sun, Certain Diagonal Equations and Conflict-Avoiding Codes of Prime Lengths, arXiv:2302.00920 [math.NT], 2023.
H. W. Lenstra, Jr., On Artin's conjecture and Euclid's algorithm in global fields, Invent. Math., Vol. 42 (1977), pp. 202-224; alternative link.
J. W. Sander, On Fibonacci primitive roots, Fibonacci Quarterly, Vol. 28, No. 1 (1990), pp. 79-80.
Daniel Shanks, Fibonacci primitive roots, end of article, Fibonacci Quarterly, Vol. 10, No. 2 (1972), pp. 163-168, 181.
Daniel Shanks and Larry Taylor, An Observation of Fibonacci Primitive Roots, Fibonacci Quarterly, Vol. 11, No. 2 (1973), pp. 159-160.
EXAMPLE
3 is a primitive root mod 5, and 3^2 = 3 + 1 mod 5, so 5 is a member. - Jonathan Sondow, Feb 17 2013
MAPLE
filter:=proc(n) local g, r;
if not isprime(n) then return false fi;
r:= [msolve(g^2 -g - 1, n)][1];
numtheory:-order(rhs(op(r)), n) = n-1
end proc:
select(filter, [5, seq(seq(10*i+j, j=[1, 9]), i=1..1000)]); # Robert Israel, May 22 2015
MATHEMATICA
okQ[p_] := AnyTrue[PrimitiveRootList[p], Mod[#^2, p] == Mod[#+1, p]&]; Select[Prime[Range[300]], okQ] (* Jean-François Alcover, Jan 04 2016 *)
PROG
(PARI) is(n)=if(kronecker(5, n)<1||!isprime(n), return(n==5)); my(s=sqrt(Mod(5, n))); znorder((1+s)/2)==n-1 || znorder((1-s)/2)==n-1 \\ Charles R Greathouse IV, May 22 2015
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from David W. Wilson
Cross-reference from Charles R Greathouse IV, Nov 05 2009
Definition clarified by M. F. Hasler, Jun 05 2018
STATUS
approved