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

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

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

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).
• Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 975.

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.
• 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)

Crossrefs

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

Programs

Magma

P := PolynomialRing(GaloisField(2)); for n := 0 to 100000 do if IsIrreducible(x^n+x+1) then print(n); end if; end for;

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 *)

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

SageMath

P. = GF(2)[]
for n in range(90):
       if (x^n+x+1).is_irreducible():
           print(n) # Ruperto Corso, Dec 11 2011

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