cp's OEIS Frontend

A308791 Primes p such that A001175(p) = (p-1)/6.

%I A308791 #13 Jul 05 2019 16:01:35
%S A308791 541,709,2389,3121,3529,4561,4861,5869,7069,8821,9001,10789,12421,
%T A308791 12781,13309,14341,14869,16981,18289,19249,19309,19429,19501,20389,
%U A308791 20809,20929,21649,22741,23629,24181,25189,26821,27109,27409,28669,30181,30469,30781,30949,31189
%N A308791 Primes p such that A001175(p) = (p-1)/6.
%C A308791 Primes p such that ord((1+sqrt(5))/2,p) = (p-1)/6, where ord(z,p) is the smallest integer k > 0 such that (z^k-1)/p is an algebraic integer.
%C A308791 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 A308791 For an odd prime p:
%C A308791 (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, ...;
%C A308791 (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, ...
%C A308791 Here k = 1, and this sequence gives primes such that (a) holds and s = 6.
%C A308791 Note that the Pisano period of {T(n)} modulo p must be even, so we have p == 1 (mod 12) for primes p in this sequence.
%C A308791 Number of terms below 10^N:
%C A308791   N | Number | Decomposing primes*
%C A308791   3 |      2 |            78
%C A308791   4 |     11 |           609
%C A308791   5 |    112 |          4777
%C A308791   6 |    898 |         39210
%C A308791   7 |   7777 |        332136
%C A308791   8 |  68115 |       2880484
%C A308791   * Here "Decomposing primes" means primes such that Legendre(5,p) = 1, i.e., p == 1, 4 (mod 5).
%t A308791 pn[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[ Fibonacci[k + 1], n] == 1, Return[k]]];
%t A308791 Reap[For[p = 2, p < 32000, p = NextPrime[p], If[Mod[p, 6] == 1, If[pn[p] == (p - 1)/6, Print[p]; Sow[p]]]]][[2, 1]] (* _Jean-François Alcover_, Jul 05 2019 *)
%o A308791 (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]))))
%o A308791 forprime(p=2, 32000, if(Pisano_for_decomposing_prime(p)==(p-1)/6, print1(p, ", ")))
%Y A308791 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), this sequence (s=6), A308792 (s=7), A308793 (s=8), A308794 (s=9).
%K A308791 nonn
%O A308791 1,1
%A A308791 _Jianing Song_, Jun 25 2019