cp's OEIS Frontend

A308801 Primes p such that A001177(p) = (p-1)/8.

%I A308801 #11 Jul 05 2019 16:02:00
%S A308801 89,761,769,1009,2089,2441,3881,4201,4289,4729,5209,5441,5849,6521,
%T A308801 6761,7369,7841,8009,8081,9929,10601,11489,11689,11801,11969,12401,
%U A308801 12409,12569,12889,14009,14249,15889,17449,17609,17881,17929,18121,18169,20201,20249,21929
%N A308801 Primes p such that A001177(p) = (p-1)/8.
%C A308801 Primes p such that ord(-(3+sqrt(5))/2,p) = (p-1)/8, where ord(z,p) is the smallest integer k > 0 such that (z^k-1)/p is an algebraic integer.
%C A308801 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 A308801 For an odd prime p:
%C A308801 (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 A308801 (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 A308801 Here k = 1, and this sequence gives primes such that (a) holds and s = 8.
%C A308801 Number of terms below 10^N:
%C A308801   N | Number | Decomposing primes*
%C A308801   3 |      3 |            78
%C A308801   4 |     20 |           609
%C A308801   5 |    154 |          4777
%C A308801   6 |   1278 |         39210
%C A308801   7 |  11063 |        332136
%C A308801   8 |  95613 |       2880484
%C A308801   * Here "Decomposing primes" means primes such that Legendre(5,p) = 1, i.e., p == 1, 4 (mod 5).
%t A308801 pn[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0, Return[k]]];
%t A308801 Reap[For[p = 2, p < 22000, p = NextPrime[p], If[Mod[p, 8] == 1, If[pn[p] == (p - 1)/8, Print[p]; Sow[p]]]]][[2, 1]] (* _Jean-François Alcover_, Jul 05 2019 *)
%o A308801 (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 A308801 forprime(p=2, 22000, if(Entry_for_decomposing_prime(p)==(p-1)/8, print1(p, ", ")))
%Y A308801 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), this sequence (s=8), A308802 (s=9).
%K A308801 nonn
%O A308801 1,1
%A A308801 _Jianing Song_, Jun 25 2019