A057461 Numbers k such that x^k + x^3 + 1 is irreducible over GF(2).

1, 2, 4, 5, 6, 7, 10, 12, 17, 18, 20, 25, 28, 31, 41, 52, 66, 130, 151, 180, 196, 503, 650, 761, 986, 1391, 1596, 2047, 2700, 4098, 6172, 6431, 6730, 8425, 10162, 11410, 12071, 13151, 14636, 17377, 18023, 30594, 32770, 65538, 77047, 81858, 102842, 130777, 137113, 143503, 168812, 192076, 262146

Offset

1,2

Comments

Next term is > 10^5. - Joerg Arndt, Apr 28 2012

It seems that if x^k + x^3 + 1 is irreducible and k is not a multiple of 6, then so is x^k + x^3 + x^2 + x + 1. If this is true, then no term can be congruent to 3 modulo 6. - Jianing Song, May 11 2021

Any subsequent terms are > 300000. - Lucas A. Brown, Nov 28 2022

Links

Table of n, a(n) for n=1..53.

Joerg Arndt, Matters Computational (The Fxtbook), section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p. 850

Lucas A. Brown, Python program.

Lucas A. Brown, Sage program.

Prog

(PARI)
for (n=1, 5000, if ( polisirreducible(Mod(1, 2)*(x^n+x^3+1)), print1(n, ", ") ) );
/* Joerg Arndt, Apr 28 2012 */
(Sage)
P.<x> = GF(2)[]
for n in range(10^4):
    if (x^n+x^3+1).is_irreducible():
        print(n) # Joerg Arndt, Apr 28 2012

Crossrefs

Cf. A002475, A057496.

Sequence in context: A129132 A191843 A318606 * A070116 A246965 A300861

Adjacent sequences: A057458 A057459 A057460 * A057462 A057463 A057464

Keyword

nonn,hard

Author

Robert G. Wilson v, Sep 27 2000

Extensions

a(24)-a(29) from Robert G. Wilson v, Aug 06 2010

Terms >= 4098 from Joerg Arndt, Apr 28 2012

a(47)-a(53) from Lucas A. Brown, Nov 28 2022

Status

approved