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).
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.
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 := 0 to 100000 do if IsIrreducible(x^n+x+1) then print(n); end if; end for;
(SageMath)
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
KEYWORD
nonn,hard,more,nice
AUTHOR
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