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

1,1

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?

Links

Robert Israel, Table of n, a(n) for n = 1..2000

Robert Israel, Linear Recurrences with a Single Minimal Period

Example

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).

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)]);

Crossrefs

Cf. A106309, A371566.

Sequence in context: A138399 A224725 A252030 * A023346 A231195 A342565

Adjacent sequences: A371566 A371567 A371568 * A371570 A371571 A371572

Keyword

nonn

Author

Robert Israel, Mar 28 2024

Status

approved