A073639 Numbers k such that x^k + x + 1 is a primitive polynomial modulo 2.

2, 3, 4, 6, 7, 15, 22, 60, 63, 127, 153, 471, 532, 865, 900, 1366

Offset: 1

Author

Richard P. Brent and Paul Zimmermann, Sep 05 2002

Keywords

• nonn
• nice
• hard
• more

Comments

Subsequence of A002475, which gives k for which the polynomial x^k + x + 1 is irreducible modulo 2. Term m of A002475 belongs to this sequence iff A046932(m) = 2^m - 1.

Note that a(16) = 1366 = A002475(23). For k = A002475(24) and A002475(25), polynomial x^k + x + 1 is not primitive modulo 2, so a(17) >= A002475(26) = 4495.

The following large terms of A002475 do not belong here: 53484, 62481, 83406, 103468. - Max Alekseyev, Aug 18 2015

Links

• Joerg Arndt, Matters Computational (The Fxtbook), section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p.850
• I. F. Blake, S. Gao and R. J. Lambert, Constructive problems for irreducible polynomials over finite fields, in Information Theory and Applications, LNCS 793, Springer-Verlag, Berlin, 1994, 1-23, See Table 2.
• R. P. Brent, Searching for primitive trinomials (mod 2)
• R. P. Brent, S. Larvala and P. Zimmermann, A fast algorithm for testing reducibility of trinomials ..., Math. Comp. 72 (2003), 1443-1452.
• N. Zierler, Primitive trinomials whose degree is a Mersenne exponent, Information and Control 15 1969 67-69.
• N. Zierler, On x^n+x+1 over GF(2), Information and Control 16 1970 502-505.
• N. Zierler and J. Brillhart, On primitive trinomials (mod 2), Information and Control 13 1968 541-554.
• N. Zierler and J. Brillhart, On primitive trinomials (mod 2), II, Information and Control 14 1969 566-569.
• Index entries for sequences related to trinomials over GF(2)

Crossrefs

Cf. A002475, A073571, A057486.

Programs

Mathematica

Select[Range[2, 1000], PrimitivePolynomialQ[x^# + x + 1, 2] &] (* Robert Price, Sep 19 2018 *)