This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A003147 Primes p with a Fibonacci primitive root g, i.e., such that g^2 = g + 1 (mod p). (Formerly M3811) 7
 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. 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 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified November 16 21:47 EST 2018. Contains 317275 sequences. (Running on oeis4.)