A132448 First primitive polynomial over GF(2) of degree n, X^n suppressed.

1, 3, 3, 3, 5, 3, 3, 29, 17, 9, 5, 83, 27, 43, 3, 45, 9, 39, 39, 9, 5, 3, 33, 27, 9, 71, 39, 9, 5, 83, 9, 175, 83, 231, 5, 119, 63, 99, 17, 57, 9, 63, 89, 101, 27, 303, 33, 183, 113, 29, 75, 9, 71, 125, 71, 149, 45, 99, 123, 3, 39, 105, 3, 27, 27, 365, 39, 163

Offset

1,2

Comments

More precisely: minimum value for X=2 of polynomials P[X] with coefficients in GF(2) such that X^n+P[X] is primitive. Applications include maximum-length linear feedback shift registers with efficient implementation in software.

Links

Francois R. Grieu, Table of n, a(n) for n = 1..400 (using Joerg Arndt's table).

Joerg Arndt, Binary primitive polynomials with lowest-most possible set bits

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

Example

a(11)=5, or 101 in binary, representing the GF(2)[X] polynomial X^2+1, because X^11+X^2+1 is primitive, contrary to X^11, X^11+1, X^11+X^1, X^11+X^1+1 and X^11+X^2.

Mathematica

i2px[i_]:=If[i>1, BitAnd[i, 1]+i2px[BitShiftRight[i, 1]]x, i ]; s={1}; For[n=2, n<69, ++n, For[i=3, !PrimitivePolynomialQ[i2px[i]+x^n, 2], i+=2]; AppendTo[s, i]]; s (* Francois R. Grieu, Jan 15 2021 *)

Crossrefs

2^n+a(n) is the smallest member of A091250 at least 2^n. A132447(n) = a(n)+2^n and gives the corresponding primitive polynomial. Cf. A132450, similar, with restriction to at most 5 terms. Cf. A132452, similar, with restriction to exactly 5 terms. Cf. A132454, similar, with restriction to minimal number of terms.

Sequence in context: A247244 A344185 A344186 * A132450 A132424 A070864

Adjacent sequences: A132445 A132446 A132447 * A132449 A132450 A132451

Keyword

nonn

Author

Francois R. Grieu, Aug 22 2007

Extensions

More terms from Francois R. Grieu, Jan 12 2021

Status

approved