A132449 First primitive GF(2)[X] polynomial of degree n with at most 5 terms.

3, 7, 11, 19, 37, 67, 131, 285, 529, 1033, 2053, 4179, 8219, 16427, 32771, 65581, 131081, 262183, 524327, 1048585, 2097157, 4194307, 8388641, 16777243, 33554441, 67108935, 134217767, 268435465, 536870917, 1073741907, 2147483657

Offset

1,1

Comments

More precisely: minimum value for X=2 of primitive GF(2)[X] polynomials of degree n with at most 5 terms. Applications include maximum-length linear feedback shift registers with efficient implementation in both hardware and software. The limitation to 5 terms occurs first for a(32), which is 4294967493 representing X^32+X^7+X^6+X^2+1, rather than 4294967471 representing X^32+X^7+X^5+X^3+X^2+X^1+1. Proof is needed that there exists a primitive GF(2)[X] polynomial P[X] of degree n and at most 5 terms for all positive n.

Links

Table of n, a(n) for n=1..31.

Index entries for sequences operating on GF(2)[X]-polynomials

Example

a(5)=37, or 100101 in binary, representing the GF(2)[X] polynomial X^5+X^2+1, because it has degree 5 and no more than 5 terms and is primitive, contrary to X^5, X^5+1, X^5+X^1, X^5+X^1+1 and X^5+X^2.

Crossrefs

Subset of A091250. A132450(n) = a(n)-2^n, giving a more compact representation. Cf. A132447, similar, with no restriction on number of terms. Cf. A132451, similar, with restriction to exactly 5 terms. Cf. A132453, similar, with restriction to minimal number of terms.

Sequence in context: A022406 A355288 A132447 * A132453 A033871 A060288

Adjacent sequences: A132446 A132447 A132448 * A132450 A132451 A132452

Keyword

nonn

Author

Francois R. Grieu, Aug 22 2007

Status

approved