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pn[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0, Return[k]]];

Reap[For[p = 2, p < 3600, p = NextPrime[p], If[pn[p] == (p - 1)/2, Print[p]; Sow[p]]]][[2, 1]] (* Jean-François Alcover, Jul 05 2019 *)

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(PARI) Entry_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])[2, 1]==0, return(v[d]))))

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For an odd rational prime p:

This Here k = 1, and this sequence gives primes such that (a) holds and s = 2. For even s, all terms are congruent to 1 modulo 4.

Number of terms below 10^N:

N | Number | Decomposing primes*

3 | 15 | 78

4 | 115 | 609

5 | 839 | 4777

6 | 6913 | 39210

7 | 58891 | 332136

8 | 510784 | 2880484

* Here "Decomposing primes" means primes such that Legendre(5,p) = 1, i.e., p == 1, 4 (mod 5).

29, 41, 101, 181, 229, 241, 349, 409, 449, 509, 569, 601, 641, 929, 941, 1021, 1061, 1109, 1129, 1201, 1229, 1321, 1481, 1489, 1549, 1609, 1621, 1669, 1709, 1741, 1789, 1801, 1861, 1889, 2029, 2069, 2129, 2609, 2621, 2861, 3209, 3301, 3361, 3389, 3449, 3461, 3581, 3761, 3769, 3821, 3889, 3989, 4129, 4229, 4241

(PARI) Entry_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])[2, 1]==0, return(v[d]))))

forprime(p=2, 3600, if(Entry_for_decomposing_prime(p)==(p-1)/2, print1(p, ", ")))

Primes p such that ord(-(13+sqrt(5))/2,p) = 2*(p+-1)/3, 2, where ord(z,p) is the smallest integer k > 0 such that (z^k-1)/p is an algebraic integer.

Primes p such that ord((1+sqrt(5))/2,p) = 2*(p+1)/3, 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 entry point Pisano period of {T(n)} modulo p (that is, the smallest m > 0 such that T(n+m) == 0 T(n) (mod p) for all n) is the multiplicative order of -ord(u^2 modulo ,p. Here ); the multiplicative order entry point of u {T(n)} modulo p (that is defined as , the smallest positive integer k m > 0 such that T(u^k-1m)/ == 0 (mod p )) is an algebraic integerord(-u^2,p).