A057461 - OEIS

%I #34 Nov 29 2022 01:19:04

%S 1,2,4,5,6,7,10,12,17,18,20,25,28,31,41,52,66,130,151,180,196,503,650,

%T 761,986,1391,1596,2047,2700,4098,6172,6431,6730,8425,10162,11410,

%U 12071,13151,14636,17377,18023,30594,32770,65538,77047,81858,102842,130777,137113,143503,168812,192076,262146

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

%C Next term is > 10^5. - _Joerg Arndt_, Apr 28 2012

%C 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

%C Any subsequent terms are > 300000. - _Lucas A. Brown_, Nov 28 2022

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p. 850

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/irred_trinom_f2.py">Python program</a>.

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/irred_trinom_f2.sage">Sage program</a>.

%o (PARI)

%o for (n=1,5000, if ( polisirreducible(Mod(1,2)*(x^n+x^3+1)), print1(n,", ") ) );

%o /* _Joerg Arndt_, Apr 28 2012 */

%o (Sage)

%o P.<x> = GF(2)[]

%o for n in range(10^4):

%o     if (x^n+x^3+1).is_irreducible():

%o         print(n) # _Joerg Arndt_, Apr 28 2012

%Y Cf. A002475, A057496.

%K nonn,hard

%O 1,2

%A _Robert G. Wilson v_, Sep 27 2000

%E a(24)-a(29) from _Robert G. Wilson v_, Aug 06 2010

%E Terms >= 4098 from _Joerg Arndt_, Apr 28 2012

%E a(47)-a(53) from _Lucas A. Brown_, Nov 28 2022