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A308797
Primes p such that A001177(p) = (p-1)/4.
8
61, 109, 149, 269, 389, 401, 701, 809, 821, 1049, 1181, 1249, 1289, 1301, 1361, 1409, 1429, 1721, 1901, 1949, 2141, 2309, 2341, 2381, 2549, 2729, 2741, 2801, 2909, 3049, 3061, 3089, 3109, 3169, 3181, 3221, 3229, 3541, 3701, 3709, 3929, 4001, 4049, 4349, 4421, 4649
OFFSET
1,1
COMMENTS
Primes p such that ord(-(3+sqrt(5))/2,p) = (p-1)/4, 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 = 4.
Number of terms below 10^N:
N | Number | Decomposing primes*
3 | 9 | 78
4 | 81 | 609
5 | 651 | 4777
6 | 5268 | 39210
7 | 44188 | 332136
8 | 383224 | 2880484
* Here "Decomposing primes" means primes such that Legendre(5,p) = 1, i.e., p == 1, 4 (mod 5).
MATHEMATICA
pn[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0, Return[k]]];
Reap[For[p = 2, p < 6000, p = NextPrime[p], If[Mod[p, 4] == 1, If[pn[p] == (p - 1)/4, Print[p]; Sow[p]]]]][[2, 1]] (* Jean-François 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, 5000, if(Entry_for_decomposing_prime(p)==(p-1)/4, print1(p, ", ")))
CROSSREFS
Similar sequences that give primes such that (a) holds: A106535 (s=1), A308795 (s=2), A308796 (s=3), this sequence (s=4), A308798 (s=5), A308799 (s=6), A308800 (s=7), A308801 (s=8), A308802 (s=9).
Sequence in context: A203263 A248431 A141919 * A155571 A107152 A141301
KEYWORD
nonn
AUTHOR
Jianing Song, Jun 25 2019
STATUS
approved