A218357 Minimal order of degree-n irreducible polynomials over GF(5).

1, 3, 31, 13, 11, 7, 19531, 32, 19, 33, 12207031, 91, 305175781, 29, 181, 17, 409, 27, 191, 41, 379, 23, 8971, 224, 101, 5227, 109, 377, 59, 61, 1861, 128, 199, 1227, 211, 37, 149, 573, 79, 241, 2238236249, 43, 1644512641, 89, 209, 47, 177635683940025046467781066894531

Offset: 1

Author

Alois P. Heinz, Oct 27 2012

Keywords

• nonn

Comments

a(n) < 5^n.

a(n) <= A143665(n). For prime n, a(n) = A143665(n). - Max Alekseyev, Apr 30 2022

Links

• Max Alekseyev, Table of n, a(n) for n = 1..520
• Eric Weisstein's World of Mathematics, Irreducible Polynomial
• Eric Weisstein's World of Mathematics, Polynomial Order

Crossrefs

Cf. A212485, A213224.

Programs

Maple

with(numtheory):
M:= proc(n) M(n):= divisors(5^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..47);

Mathematica

M[n_] := M[n] = Divisors[5^n - 1] ~Complement~ U[n-1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 47}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Formula

a(n) = min(M(n)) with M(n) = {d : d|(5^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A212485(n,1) = A213224(n,3).