A057486 Numbers k such that x^k + x^m + 1 is factorable over GF(2) for all m between 1 and k.

8, 13, 16, 19, 24, 26, 27, 32, 37, 38, 40, 43, 45, 48, 50, 51, 53, 56, 59, 61, 64, 67, 69, 70, 72, 75, 77, 78, 80, 82, 83, 85, 88, 91, 96, 99, 101, 104, 107, 109, 112, 114, 115, 116, 117, 120, 122, 125, 128, 131, 133, 136, 138, 139, 141, 143, 144, 149, 152, 157

Offset: 1

Author

Robert G. Wilson v, Sep 28 2000

Keywords

• nonn

Comments

Brent, Hart, Kruppa, and Zimmermann found that 57885161 is a term of this sequence. - Charles R Greathouse IV, May 30 2013

Examples

			a(1) = 8 because
x^8 + x^1 + 1 = (1 + x + x^2)*(1 + x^2 + x^3 + x^5 + x^6),
x^8 + x^2 + 1 = (1 + x + x^4)^2,
x^8 + x^3 + 1 = (1 + x + x^3)*(1 + x + x^2 + x^3 + x^5),
x^8 + x^4 + 1 = (1 + x + x^2)^4,
x^8 + x^5 + 1 = (1 + x^2 + x^3)*(1 + x^2 + x^3 + x^4 + x^5),
x^8 + x^6 + 1 = (1 + x^3 + x^4)^2, and
x^8 + x^7 + 1 = (1 + x + x^2)*(1 + x + x^3 + x^4 + x^6).
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..5000 (first 200 terms from T. D. Noe)
• Richard Brent, The Software gf2x.
• Paul Zimmermann, There is no primitive trinomial of degree 57885161 over GF(2), posting to NMBRTHRY mailing list [alternate link]
• Index entries for sequences related to trinomials over GF(2)

Crossrefs

Complement of A073571. Cf. A001153, A002475, A073639.

Programs

Mathematica

Do[ k = 1; While[ ToString[ Factor[ x^n + x^k + 1, Modulus -> 2 ]] != ToString[ x^n + x^k + 1 ] && k < n, k++ ]; If[ k == n, Print[ n ]], {n, 2, 234} ]

PARI

is(n)=for(s=1,n\2,if(polisirreducible((x^n+x^s+1)*Mod(1,2)), return(0)));1 \\ Charles R Greathouse IV, May 30 2013