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A308794 Primes p such that A001175(p) = (p-1)/9. 7
199, 919, 6679, 7489, 12979, 16921, 17011, 17659, 20089, 20431, 23059, 23599, 24391, 24859, 25309, 28081, 29629, 33301, 36901, 39079, 39439, 41761, 42589, 43399, 43669, 45361, 46261, 48619, 51481, 53479, 54091, 62011, 62191, 67411, 69499, 72019, 72091, 77419, 78301 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p such that ord((1+sqrt(5))/2,p) = (p-1)/9, where ord(z,p) is the smallest integer k > 0 such that (z^k-1)/p is an algebraic integer.

Let {T(n)} be a sequence defined by T(0) = 0, T(1) = 1, T(n) = k*T(n-1) + T(n-2), K be the quadratic field Q[sqrt(k^2+4)], O_K be the ring of integer of K, u = (k+sqrt(k^2+4))/2. For a prime p not dividing k^2 + 4, the Pisano period of {T(n)} modulo p (that is, the smallest m > 0 such that T(n+m) == T(n) (mod p) for all n) is ord(u,p); the entry point of {T(n)} modulo p (that is, the smallest m > 0 such that T(m) == 0 (mod p)) is ord(-u^2,p).

For an odd prime p:

(a) if p decomposes in K, then (O_K/pO_K)* (the multiplicative group of O_K modulo p) is congruent to C_(p-1) X C_(p-1), so the Pisano period of {T(n)} modulo p is equal to (p-1)/s, s = 1, 2, 3, 4, ...;

(b) if p is inert in K, then u^(p+1) == -1 (mod p), so the Pisano period of {T(n)} modulo p is equal to 2*(p+1)/r, r = 1, 3, 5, 7, ...

Here k = 1, and this sequence gives primes such that (a) holds and s = 9.

Number of terms below 10^N:

  N | Number | Decomposing primes*

  3 |      2 |            78

  4 |      4 |           609

  5 |     49 |          4777

  6 |    405 |         39210

  7 |   3489 |        332136

  8 |  30132 |       2880484

  * Here "Decomposing primes" means primes such that Legendre(5,p) = 1, i.e., p == 1, 4 (mod 5).

LINKS

Table of n, a(n) for n=1..39.

MATHEMATICA

pn[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[ Fibonacci[k + 1], n] == 1, Return[k]]];

Reap[For[p = 2, p < 50000, p = NextPrime[p], If[Mod[p, 9] == 1, If[pn[p] == (p - 1)/9, Print[p]; Sow[p]]]]][[2, 1]] (* Jean-Fran├žois Alcover, Jul 05 2019 *)

PROG

(PARI) Pisano_for_decomposing_prime(p) = my(k=1, M=[k, 1; 1, 0], Id=[1, 0; 0, 1]); if(isprime(p)&&kronecker(k^2+4, p)==1, my(v=divisors(p-1)); for(d=1, #v, if(Mod(M, p)^v[d]==Id, return(v[d]))))

forprime(p=2, 80000, if(Pisano_for_decomposing_prime(p)==(p-1)/9, print1(p, ", ")))

CROSSREFS

Similar sequences that give primes such that (a) holds: A003147/{5} (s=1), A308787 (s=2), A308788 (s=3), A308789 (s=4), A308790 (s=5), A308791 (s=6), A308792 (s=7), A308793 (s=8), this sequence (s=9).

Sequence in context: A105975 A095995 A159657 * A308802 A142570 A155507

Adjacent sequences:  A308791 A308792 A308793 * A308795 A308796 A308797

KEYWORD

nonn

AUTHOR

Jianing Song, Jun 25 2019

STATUS

approved

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Last modified September 22 05:47 EDT 2019. Contains 327287 sequences. (Running on oeis4.)