

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
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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) = n1.  Matthew Goers, Sep 20 2013
As shown in the paper by Brison, these are also the primes p such that there is a Fibonaccitype sequence (mod p) that begins with (1,b) and encounters all numbers less than p in the first p1 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. 295304.
Alexandru Gica, Quadratic Residues in Fibonacci Sequences, Fibonacci Quart. 46/47 (2008/2009), no. 1, 6872. See Theorem 5.1.
D. Shanks, Fibonacci primitive roots, end of article, Fib. Quart., 10 (1972), 163168, 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) = n1
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] (* JeanFrançois 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)==n1  znorder((1s)/2)==n1 \\ 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
Crossreference from Charles R Greathouse IV, Nov 05 2009
Definition clarified by M. F. Hasler, Jun 05 2018


STATUS

approved



