A132451 First primitive GF(2)[X] polynomials of degree n with exactly 5 terms.

0, 0, 0, 0, 47, 91, 143, 285, 539, 1051, 2071, 4179, 8219, 16427, 32791, 65581, 131087, 262183, 524327, 1048659, 2097191, 4194361, 8388651, 16777243, 33554447, 67108935, 134217767, 268435539, 536870935, 1073741907, 2147483663

Offset: 1

Author

Francois R. Grieu (f(AT)grieu.com), Aug 22 2007

Keywords

• nonn

Comments

More precisely: minimum value for X=2 of primitive GF(2)[X] polynomials of degree n with exactly 5 terms, or 0 if no such polynomial exists. Applications include maximum-length linear feedback shift registers with efficient implementation in both hardware and software. 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.

Examples

			a(6)=91, or 1011011 in binary, representing the GF(2)[X] polynomial X^6+X^4+X^3+X^1+1, because it has degree 6 and exactly 5 terms and is primitive, contrary to X^6+X^3+X^2+X^1+1 and X^6+X^4+X^2+X^1+1.
		

Links

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

Crossrefs

For n>4, a(n) belongs to A091250. A132452(n) = a(n)-2^n, giving a more compact representation. Cf. A132447, similar, with no restriction on number of terms. Cf. A132449, similar, with restriction to a most 5 terms. Cf. A132453, similar, with restriction to minimal number of terms.