cp's OEIS Frontend

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

%I A057461 #34 Nov 29 2022 01:19:04
%S A057461 1,2,4,5,6,7,10,12,17,18,20,25,28,31,41,52,66,130,151,180,196,503,650,
%T A057461 761,986,1391,1596,2047,2700,4098,6172,6431,6730,8425,10162,11410,
%U A057461 12071,13151,14636,17377,18023,30594,32770,65538,77047,81858,102842,130777,137113,143503,168812,192076,262146
%N A057461 Numbers k such that x^k + x^3 + 1 is irreducible over GF(2).
%C A057461 Next term is > 10^5. - _Joerg Arndt_, Apr 28 2012
%C A057461 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 A057461 Any subsequent terms are > 300000. - _Lucas A. Brown_, Nov 28 2022
%H A057461 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 A057461 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/irred_trinom_f2.py">Python program</a>.
%H A057461 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/irred_trinom_f2.sage">Sage program</a>.
%o A057461 (PARI)
%o A057461 for (n=1,5000, if ( polisirreducible(Mod(1,2)*(x^n+x^3+1)), print1(n,", ") ) );
%o A057461 /* _Joerg Arndt_, Apr 28 2012 */
%o A057461 (Sage)
%o A057461 P.<x> = GF(2)[]
%o A057461 for n in range(10^4):
%o A057461     if (x^n+x^3+1).is_irreducible():
%o A057461         print(n) # _Joerg Arndt_, Apr 28 2012
%Y A057461 Cf. A002475, A057496.
%K A057461 nonn,hard
%O A057461 1,2
%A A057461 _Robert G. Wilson v_, Sep 27 2000
%E A057461 a(24)-a(29) from _Robert G. Wilson v_, Aug 06 2010
%E A057461 Terms >= 4098 from _Joerg Arndt_, Apr 28 2012
%E A057461 a(47)-a(53) from _Lucas A. Brown_, Nov 28 2022