A057496 Numbers n such that x^n + x^3 + x^2 + x + 1 is irreducible over GF(2).

1, 2, 3, 4, 5, 7, 10, 17, 20, 25, 28, 31, 41, 52, 130, 151, 196, 503, 650, 761, 986, 1391, 2047, 6172, 6431, 6730, 8425, 10162, 11410, 12071, 13151, 14636, 17377, 18023, 32770, 77047, 102842, 130777, 137113, 143503, 168812, 192076

Offset

1,2

Comments

If x^n + x^3 + x^2 + x + 1 is irreducible, then so is its "twin" x^n + x^3 + 1. - Gove Effinger, Mar 11 2007

No term other than 3 can be a multiple of 3, since for m > 1, x^(3*m) + x^3 + x^2 + x + 1 is divisible by x^2 + x + 1. - Jianing Song, May 11 2021

Links

Table of n, a(n) for n=1..42.

Formula

Using probabilistic arguments it appears that there should be about 6.5 terms in this sequence with any given number of decimal digits d. - Gove Effinger, Mar 11 2007

Crossrefs

Cf. A002475.

Other than the term 3, subsequence of A057461.

Sequence in context: A282502 A212463 A130080 * A345663 A191864 A180348

Adjacent sequences: A057493 A057494 A057495 * A057497 A057498 A057499

Keyword

nonn

Author

Robert G. Wilson v, Sep 27 2000

Extensions

a(20)-a(23) from Robert G. Wilson v, Mar 11 2007

a(24)-a(27) computed by Richard P. Brent, Mar 11 2007, communicated by Gove Effinger

a(27)-a(35) computed by Richard P. Brent, Mar 16 2007, communicated by Gove Effinger

a(36)-a(42) computed by Jonathan Webster, Feb 18 2010

Added entries a(1), a(2), a(3) since x^3 + x^2 + 1, x^3 + x + 1 and x^2 + x + 1 are irreducible over GF(2). Changed the offset for the entries computed by Robert G. Wilson v and Richard P. Brent to account for this. Added terms a(36) through a(42). - Jonathan Webster (jwebster(AT)bates.edu), Feb 18 2010

Status

approved