login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A073639 Numbers n such that x^n + x + 1 is a primitive polynomial modulo 2. 6
2, 3, 4, 6, 7, 15, 22, 60, 63, 127, 153, 471, 532, 865, 900, 1366 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Subsequence of A002475 that gives n for which the polynomial x^n + 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 n = A002475(24) and A002475(25), polynomial x^n + 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

Table of n, a(n) for n=1..16.

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)

MATHEMATICA

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

CROSSREFS

Cf. A002475, A073571, A057486.

Sequence in context: A039059 A151892 A162570 * A130776 A077292 A270475

Adjacent sequences:  A073636 A073637 A073638 * A073640 A073641 A073642

KEYWORD

nonn,nice,hard,more

AUTHOR

Richard P. Brent and Paul Zimmermann, Sep 05 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 19 12:31 EST 2019. Contains 320310 sequences. (Running on oeis4.)