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

Author

Robert G. Wilson v, Sep 27 2000

Keywords

• nonn
• hard

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

• 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.

Crossrefs

Cf. A002475, A057496.

Programs

PARI

for (n=1,5000, if ( polisirreducible(Mod(1,2)*(x^n+x^3+1)), print1(n,", ") ) );
/* Joerg Arndt, Apr 28 2012 */

Sage

P. = GF(2)[]
for n in range(10^4):
    if (x^n+x^3+1).is_irreducible():
        print(n) # Joerg Arndt, Apr 28 2012

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