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A308802 Primes p such that A001177(p) = (p-1)/9. 8
199, 919, 6679, 12979, 17011, 17659, 20431, 23059, 23599, 24391, 24859, 39079, 39439, 43399, 48619, 53479, 54091, 62011, 62191, 67411, 69499, 72019, 72091, 77419, 81019, 82279, 91099, 91459, 92179, 97579, 98731, 102259, 103231, 105211, 108271, 111439, 114679, 125119 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p such that ord(-(3+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 entry point 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), (-u^2)^(p+1) == 1 (mod p), so the entry point of {T(n)} modulo p is equal to (p+1)/s, s = 1, 2, 3, 4, ...

Here k = 1, and this sequence gives primes such that (a) holds and s = 9. For odd s, all terms are congruent to 3 modulo 4.

Number of terms below 10^N:

  N | Number | Decomposing primes*

  3 |      2 |            78

  4 |      3 |           609

  5 |     31 |          4777

  6 |    274 |         39210

  7 |   2293 |        332136

  8 |  20097 |       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..38.

MATHEMATICA

pn[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0, 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) Entry_for_decomposing_prime(p) = my(k=1, M=[k, 1; 1, 0]); if(isprime(p)&&kronecker(k^2+4, p)==1, my(v=divisors(p-1)); for(d=1, #v, if((Mod(M, p)^v[d])[2, 1]==0, return(v[d]))))

forprime(p=2, 126000, if(Entry_for_decomposing_prime(p)==(p-1)/9, print1(p, ", ")))

CROSSREFS

Similar sequences that give primes such that (a) holds: A106535 (s=1), A308795 (s=2), A308796 (s=3), A308797 (s=4), A308798 (s=5), A308799 (s=6), A308800 (s=7), A308801 (s=8), this sequence (s=9).

Sequence in context: A095995 A159657 A308794 * A142570 A155507 A159064

Adjacent sequences:  A308799 A308800 A308801 * A308803 A308804 A308805

KEYWORD

nonn

AUTHOR

Jianing Song, Jun 25 2019

STATUS

approved

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Last modified September 17 06:41 EDT 2019. Contains 327119 sequences. (Running on oeis4.)