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A003147 Primes p with a Fibonacci primitive root g, i.e., such that g^2 = g + 1 (mod p).
(Formerly M3811)
15
5, 11, 19, 31, 41, 59, 61, 71, 79, 109, 131, 149, 179, 191, 239, 241, 251, 269, 271, 311, 359, 379, 389, 409, 419, 431, 439, 449, 479, 491, 499, 569, 571, 599, 601, 631, 641, 659, 701, 719, 739, 751, 821, 839, 929, 971, 1019, 1039, 1051, 1091, 1129, 1171, 1181, 1201, 1259, 1301 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Noe)

Owen J. Brison, Complete Fibonacci sequences in finite fields, Fibonacci Quarterly, 30 (1992), pp. 295-304.

Alexandru Gica, Quadratic Residues in Fibonacci Sequences, Fibonacci Quart. 46/47 (2008/2009), no. 1, 68-72. See Theorem 5.1.

D. Shanks, Fibonacci primitive roots, end of article, Fib. Quart., 10 (1972), 163-168, 181.

Index entries for primes by primitive root

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-Fran├žois 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

Subsequence of A038872. Cf. A001175

Sequence in context: A274946 A253936 A191032 * A106068 A304875 A164566

Adjacent sequences:  A003144 A003145 A003146 * A003148 A003149 A003150

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

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

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Last modified December 7 17:41 EST 2019. Contains 329847 sequences. (Running on oeis4.)