cp's OEIS Frontend

A132452 First primitive GF(2)[X] polynomials of degree n with exactly 5 terms, X^n suppressed.

%I A132452 #19 May 06 2022 13:13:51
%S A132452 15,27,15,29,27,27,23,83,27,43,23,45,15,39,39,83,39,57,43,27,15,71,39,
%T A132452 83,23,83,15,197,83,281,387,387,83,99,147,57,15,153,89,101,27,449,51,
%U A132452 657,113,29,75,75,71,329,71,149,45,99,149,53,39,105,51,27,27,833,39,163,101,43,43,1545,29
%N A132452 First primitive GF(2)[X] polynomials of degree n with exactly 5 terms, X^n suppressed.
%C A132452 More precisely: minimum value for X=2 of GF(2)[X] polynomials P[X] of degree less than n and exactly 4 terms such that X^n+P[X] is primitive.
%C A132452 Applications include maximum-length linear feedback shift registers with efficient implementation in both hardware and software.
%C A132452 Proof is needed that there exists a primitive GF(2)[X] polynomial P[X] of degree n and exactly 5 terms for all n>4.
%H A132452 Joerg Arndt, <a href="/A132452/b132452.txt">Table of n, a(n) for n = 5..400</a>
%H A132452 <a href="/index/Ge#GF2X">Index entries for sequences operating on GF(2)[X]-polynomials</a>
%e A132452 a(11)=23, or 10111 in binary, representing the GF(2)[X] polynomial X^4+X^2+X^1+1, because X^11+X^4+X^2+X^1+1 has exactly 5 terms and it is primitive, contrary to X^11+X^3+X^2+X^1+1.
%Y A132452 For n>4, 2^n+a(n) belongs to A091250. A132451(n) = a(n)+2^n and gives the corresponding primitive polynomial. Cf. A132448, similar, with no restriction on number of terms. Cf. A132450, similar, with restriction to at most 5 terms. Cf. A132454, similar, with restriction to minimal number of terms.
%K A132452 nonn
%O A132452 5,1
%A A132452 _Francois R. Grieu_, Aug 22 2007
%E A132452 Edited and extended by _Max Alekseyev_, Feb 06 2010