A327753 Primes powers (A246655) congruent to 4 mod 5.

4, 9, 19, 29, 49, 59, 64, 79, 89, 109, 139, 149, 169, 179, 199, 229, 239, 269, 289, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, 509, 529, 569, 599, 619, 659, 709, 719, 729, 739, 769, 809, 829, 839, 859, 919, 929, 1009, 1019, 1024, 1039, 1049, 1069, 1109, 1129, 1229, 1249

Offset: 1

Author

Jianing Song, Sep 24 2019

Keywords

• nonn

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),

Examples

			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).
		

Links

• Marius A. Burtea, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A137827, A327752.

Intersection of A016897 and A246655.

Programs

Magma

[n:n in [2..1250]|IsPrimePower(n) and (n mod 5 eq 4)]; // Marius A. Burtea, Sep 26 2019

Mathematica

Select[Range@ 1250, And[PrimePowerQ@ #, Mod[#, 5] == 4] &] (* Michael De Vlieger, Sep 27 2019 *)

PARI

isok(n) = isprimepower(n) && (n%5==4)