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

1,1

Comments

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

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)

Example

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).

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} ]

Prog

(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

Crossrefs

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

Sequence in context: A328241 A287159 A272486 * A188198 A129410 A338598

Adjacent sequences: A057483 A057484 A057485 * A057487 A057488 A057489

Keyword

nonn

Author

Robert G. Wilson v, Sep 28 2000

Status

approved