A057463 Numbers k such that x^k + x^4 + 1 is irreducible over GF(2).

1, 3, 7, 9, 15, 39, 57, 81, 105, 1239, 5569, 9457, 11095, 11631, 12327, 37633, 63247, 216457

Offset: 1

Author

Robert G. Wilson v, Sep 27 2000

Keywords

• nonn
• more

Comments

a(18) is greater than 10^5. - Joerg Arndt, Apr 28 2012

All terms are congruent to 1 or 3 (mod 6). - Robert Israel, Sep 05 2016

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

Examples

			6 is not in the sequence since x^6 + x^4 + 1 = (x^3 + x^2 + 1)^2, but 7 is in the sequence since x^7 + x^4 + 1 is irreducible. (Trial division by x + 1, x^2 + x + 1, x^3 + x^2 + 1, and x^3 + x + 1) - _Michael B. Porter_, Sep 06 2016
		

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.

Crossrefs

Cf. A002475.

Programs

Maple

for m from 1 to 200 do if(Irreduc(x^m + x^4 + 1) mod 2) then printf("%d, ",m):fi:od: # Nathaniel Johnston, Apr 19 2011

Sage

P. = GF(2)[]
for n in range(10^4):
    if (x^n+x^4+1).is_irreducible():
        print(n) # Joerg Arndt, Apr 28 2012

Extensions

a(10)-a(15) from Nathaniel Johnston, Apr 19 2011

a(16)-a(17) from Joerg Arndt, Apr 28 2012

a(18) from Lucas A. Brown, Nov 28 2022