A327752 Primes powers (A246655) congruent to 1 mod 5.

11, 16, 31, 41, 61, 71, 81, 101, 121, 131, 151, 181, 191, 211, 241, 251, 256, 271, 281, 311, 331, 361, 401, 421, 431, 461, 491, 521, 541, 571, 601, 631, 641, 661, 691, 701, 751, 761, 811, 821, 841, 881, 911, 941, 961, 971, 991, 1021, 1031, 1051, 1061, 1091, 1151, 1171, 1181, 1201

Offset: 1

Author

Jianing Song, Sep 24 2019

Keywords

• nonn

Comments

Numbers k, not powers of 5, such that x^4 + x^3 + x^2 + x + 1 factors into four linear polynomials over GF(k).

This sequence consists of numbers of the form p^e where prime p == 1 (mod 5), p^(2e) where prime p == 4 (mod 5) and p^(4e) where prime p == 2, 3 (mod 5),

Examples

			k = 11: in GF(11), x^4 + x^3 + x^2 + x + 1 = (x - 3)*(x - 4)*(x - 5)*(x + 2);
k = 16: let GF(16) = GF(2)[y]/(y^4+y+1), then x^4 + x^3 + x^2 + x + 1 = (x - y^3)*(x - (y^3+y))*(x - (y^3+y^2))*(x - (y^3+y^2+y+1)).
		

Links

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

Crossrefs

Cf. A137827, A327753.

Intersection of A016861 and A246655.

Programs

Magma

[n:n in [2..1210]|IsPrimePower(n) and (n mod 5 eq 1)]; // Marius A. Burtea, Sep 26 2019

PARI

isok(n) = isprimepower(n) && (n%5==1)