cp's OEIS Frontend

A308802 Primes p such that A001177(p) = (p-1)/9.

%I A308802 #11 Jul 05 2019 16:02:03
%S A308802 199,919,6679,12979,17011,17659,20431,23059,23599,24391,24859,39079,
%T A308802 39439,43399,48619,53479,54091,62011,62191,67411,69499,72019,72091,
%U A308802 77419,81019,82279,91099,91459,92179,97579,98731,102259,103231,105211,108271,111439,114679,125119
%N A308802 Primes p such that A001177(p) = (p-1)/9.
%C A308802 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.
%C A308802 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).
%C A308802 For an odd prime p:
%C A308802 (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, ...;
%C A308802 (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, ...
%C A308802 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.
%C A308802 Number of terms below 10^N:
%C A308802   N | Number | Decomposing primes*
%C A308802   3 |      2 |            78
%C A308802   4 |      3 |           609
%C A308802   5 |     31 |          4777
%C A308802   6 |    274 |         39210
%C A308802   7 |   2293 |        332136
%C A308802   8 |  20097 |       2880484
%C A308802   * Here "Decomposing primes" means primes such that Legendre(5,p) = 1, i.e., p == 1, 4 (mod 5).
%t A308802 pn[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0, Return[k]]];
%t A308802 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-François Alcover_, Jul 05 2019 *)
%o A308802 (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]))))
%o A308802 forprime(p=2, 126000, if(Entry_for_decomposing_prime(p)==(p-1)/9, print1(p, ", ")))
%Y A308802 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).
%K A308802 nonn
%O A308802 1,1
%A A308802 _Jianing Song_, Jun 25 2019