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

15, 27, 15, 29, 27, 27, 23, 83, 27, 43, 23, 45, 15, 39, 39, 83, 39, 57, 43, 27, 15, 71, 39, 83, 23, 83, 15, 197, 83, 281, 387, 387, 83, 99, 147, 57, 15, 153, 89, 101, 27, 449, 51, 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

Offset: 5

Author

Francois R. Grieu, Aug 22 2007

Keywords

• nonn

Comments

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.

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(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.
		

Links

• Joerg Arndt, Table of n, a(n) for n = 5..400
• Index entries for sequences operating on GF(2)[X]-polynomials

Crossrefs

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.

Extensions

Edited and extended by Max Alekseyev, Feb 06 2010