A371569 Primes p such that for all initial conditions (x(0),x(1),x(2),x(3),x(4)) in [0..p-1]^5 except [0,0,0,0,0], the 5-step recurrence x(k) = x(k-1) + x(k-2) + x(k-3) + x(k-4) + x(k-5) (mod p) has the same period, but x^5 - x^4 - x^3 - x^2 - x - 1 is reducible (mod p).

4259, 61643, 94307, 110063, 118171, 348149, 1037903, 1872587, 2149403, 2331859, 2450807, 2490263, 2500847, 2521823, 2534659, 2772179, 2788367, 2789939, 3271883, 3399707, 3550751, 3577487, 3640859, 3861899, 3904309, 4016219, 4063211, 4236719, 4245239, 4368739, 4441007, 4542779, 5033477, 5446283

Offset: 1

Author

Robert Israel, Mar 28 2024

Keywords

• nonn

Comments

Terms of A106309 that are not in A371566.

In each of the first 2000 terms, x^5 - x^4 - x^3 - x^2 - x - 1 splits into linear factors (mod p). Are there any where it does not?

Examples

			a(3) = 94307 is a term because 94307 is prime, z^5  - z^4 - z^3 - z^2 - z - 1 = (z + 11827)*(z + 78583)*(z + 54610)*(z + 14536)*(z + 29057) (mod 94307), and the recurrence has period 47153 for all initial conditions except (0,0,0,0,0), as -11827, -78583, -54610, -14536, and -29057 all have multiplicative order 47153 (mod 94307).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2000
• Robert Israel, Linear Recurrences with a Single Minimal Period

Crossrefs

Cf. A106309, A371566.

Programs

Maple

filter:= proc(p) local Q, q, F, i, z, d, k, kp, G, alpha;
  if not isprime(p) then return false fi;
  Q:= z^5  - z^4 - z^3 - z^2 - z - 1;
  if Irreduc(Q) mod p then return false fi;
  F:= (Factors(Q) mod p)[2];
  if ormap(t -> t[2]>1, F) then return false fi;
  for i from 1 to nops(F) do
     q:= F[i][1];
     d:= degree(q);
     if d = 1 then kp:= NumberTheory:-MultiplicativeOrder(p+solve(q, z), p);
     else
         G:= GF(p, d, q);
         alpha:= G:-ConvertIn(z);
         kp:= G:-order(alpha);
     fi;
     if i = 1 then k:= kp
     elif kp <> k then return false
     fi;
  od;
  true
end proc:
select(filter, [seq(i, i=3 .. 10^7,2)]);