OFFSET
1,1
COMMENTS
Numbers k such that x^4 + x^3 + x^2 + x + 1 factors into two irreducible quadratic polynomials over GF(k).
Note that x^4 + x^3 + x^2 + x + 1 is reducible over GF(k) if and only if there exists some a in GF(k) such that a^2 - a - 1 = 0, and then x^4 + x^3 + x^2 + x + 1 = (x^2 + a*x + 1) * (x^2 + (1-a)*x + 1). There exists some a in GF(k) such that a^2 - a - 1 = 0 if and only if kronecker(k,5) = 1, or k == 1, 4 (mod 5). If k == 1 (mod 5), then x^4 + x^3 + x^2 + x + 1 can be further factored into four linear polynomials.
This sequence consists of numbers of the form p^(2e+1) where prime p == 4 (mod 5) and p^(4e+2) where prime p == 2, 3 (mod 5),
LINKS
Marius A. Burtea, Table of n, a(n) for n = 1..10000
EXAMPLE
k = 4: let GF(4) = GF(2)[w], w^2 + w + 1 = 0, then x^4 + x^3 + x^2 + x + 1 = (x^2 + w*x + 1)*(x^2 + (w+1)*x + 1);
k = 9: let GF(9) = GF(3)[i], i^2 = -1, then x^4 + x^3 + x^2 + x + 1 = (x^2 + (-1+i)*x + 1)*(x^2 + (-1-i)*x + 1);
k = 19: in GF(19), x^4 + x^3 + x^2 + x + 1 = (x^2 + 5x + 1)*(x^2 - 4x + 1).
MATHEMATICA
Select[Range@ 1250, And[PrimePowerQ@ #, Mod[#, 5] == 4] &] (* Michael De Vlieger, Sep 27 2019 *)
PROG
(PARI) isok(n) = isprimepower(n) && (n%5==4)
(Magma) [n:n in [2..1250]|IsPrimePower(n) and (n mod 5 eq 4)]; // Marius A. Burtea, Sep 26 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 24 2019
STATUS
approved