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A308792 Primes p such that A001175(p) = (p-1)/7. 8
2269, 2731, 2969, 3739, 4831, 6091, 6329, 11159, 11789, 13049, 13679, 14281, 14449, 14771, 16871, 19559, 20399, 24179, 26111, 29191, 31039, 33181, 33811, 34511, 34679, 35911, 40111, 41651, 42701, 43961, 49211, 54881, 55259, 55721, 56099, 58129, 60859, 62819, 66809 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p such that ord((1+sqrt(5))/2,p) = (p-1)/7, 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 = 7.

Number of terms below 10^N:

  N | Number | Decomposing primes*

  3 |      0 |            78

  4 |      7 |           609

  5 |     55 |          4777

  6 |    507 |         39210

  7 |   4144 |        332136

  8 |  36319 |       2880484

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

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

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, 7] == 1, If[pn[p] == (p - 1)/7, 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, 67000, if(Pisano_for_decomposing_prime(p)==(p-1)/7, 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), this sequence (s=7), A308793 (s=8), A308794 (s=9).

Sequence in context: A205756 A205586 A125253 * A236606 A251493 A235519

Adjacent sequences:  A308789 A308790 A308791 * A308793 A308794 A308795

KEYWORD

nonn

AUTHOR

Jianing Song, Jun 25 2019

STATUS

approved

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Last modified May 28 07:06 EDT 2022. Contains 354112 sequences. (Running on oeis4.)