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A003147 Primes with a Fibonacci primitive root.
(Formerly M3811)
5
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 A164566 A075322

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

STATUS

approved

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Last modified September 25 12:56 EDT 2017. Contains 292469 sequences.