A002475 Numbers k such that x^k + x + 1 is irreducible over GF(2). (Formerly M0544 N0194)

0, 2, 3, 4, 6, 7, 9, 15, 22, 28, 30, 46, 60, 63, 127, 153, 172, 303, 471, 532, 865, 900, 1366, 2380, 3310, 4495, 6321, 7447, 10198, 11425, 21846, 24369, 27286, 28713, 32767, 34353, 46383, 53484, 62481, 83406, 87382, 103468, 198958, 248833

Offset

1,2

Comments

k=1 is excluded since the polynomial "1" is not normally regarded as irreducible.

2^(A073639(m)) - 1 is a term for all m. - Joerg Arndt, Aug 23 2015

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

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 975.

Links

Table of n, a(n) for n=1..44.

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.

N. Zierler, On x^n+x+1 over GF(2), Information and Control, 16 1970 502-505.

Index entries for sequences related to trinomials over GF(2)

Maple

select(n -> Irreduc(x^n+x+1) mod 2, [0, $2..10000]); # Robert Israel, Aug 09 2015

Mathematica

Do[ If[ ToString[ Factor[ x^n + x + 1, Modulus -> 2 ] ] == ToString[ x^n + x + 1 ], Print [ n ] ], {n, 0, 28713} ]
Select[Range[1000], IrreduciblePolynomialQ[x^# + x + 1, Modulus -> 2] &] (* Robert Price, Sep 19 2018 *)

Prog

(Magma) P<x> := PolynomialRing(GaloisField(2)); for n := 2 to 100000 do if IsIrreducible(x^n+x+1) then print(n); end if; endfor;
(Sage)
P.<x> = GF(2)[]
for n in range(90):
       if (x^n+x+1).is_irreducible():
           print(n) # Ruperto Corso, Dec 11 2011
(PARI)
for (n=1, 10^6, if ( polisirreducible(Mod(1, 2)*(x^n+x+1)), print1(n, ", ") ) );
/* Joerg Arndt, Apr 28 2012 */
(PARI) is(n)=if(n>3&&[1, 0, 1, 1, 0, 1, 0, 0][n%8+1], return(0)); polisirreducible(Mod('x^n+'x+1, 2)) \\ Charles R Greathouse IV, Jun 04 2015

Crossrefs

Cf. A001153, A073639, A057496, A223938 (n such that x^n-x-1 is irreducible over GF(3)).

Sequence in context: A165773 A064414 A224482 * A208281 A306074 A250252

Adjacent sequences: A002472 A002473 A002474 * A002476 A002477 A002478

Keyword

nonn,hard,more,nice

Author

N. J. A. Sloane

Extensions

Two more terms from Paul Zimmermann, Sep 05 2002

a(37)-a(39) from Max Alekseyev, Oct 29 2011

a(40)-a(41) from Ruperto Corso, Dec 11 2011

a(42) from Manfred Scheucher, Jun 04 2015

a(43) from Manfred Scheucher, Aug 09 2015

a(44) from Lucas A. Brown, Nov 28 2022

Status

approved