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A073639 Numbers k such that x^k + x + 1 is a primitive polynomial modulo 2. 6

%I #38 Jul 16 2021 13:17:54

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

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

%C 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.

%C 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.

%C The following large terms of A002475 do not belong here: 53484, 62481, 83406, 103468. - _Max Alekseyev_, Aug 18 2015

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 40.9.3 "Irreducible trinomials of the form 1 + x^k + x^d", p.850

%H I. F. Blake, S. Gao and R. J. Lambert, <a href="http://dx.doi.org/10.1007/3-540-57936-2_27">Constructive problems for irreducible polynomials over finite fields</a>, in Information Theory and Applications, LNCS 793, Springer-Verlag, Berlin, 1994, 1-23, See Table 2.

%H R. P. Brent, <a href="http://wwwmaths.anu.edu.au/~brent/trinom-old.html">Searching for primitive trinomials (mod 2)</a>

%H R. P. Brent, S. Larvala and P. Zimmermann, <a href="http://wwwmaths.anu.edu.au/~brent/pd/rpb199.pdf">A fast algorithm for testing reducibility of trinomials ...</a>, Math. Comp. 72 (2003), 1443-1452.

%H N. Zierler, <a href="http://dx.doi.org/10.1016/S0019-9958(69)90631-7">Primitive trinomials whose degree is a Mersenne exponent</a>, Information and Control 15 1969 67-69.

%H N. Zierler, <a href="http://dx.doi.org/10.1016/S0019-9958(70)90264-0">On x^n+x+1 over GF(2)</a>, Information and Control 16 1970 502-505.

%H N. Zierler and J. Brillhart, <a href="http://dx.doi.org/10.1016/S0019-9958(68)90973-X">On primitive trinomials (mod 2)</a>, Information and Control 13 1968 541-554.

%H N. Zierler and J. Brillhart, <a href="http://dx.doi.org/10.1016/S0019-9958(69)90356-8">On primitive trinomials (mod 2), II</a>, Information and Control 14 1969 566-569.

%H <a href="/index/Tri#trinomial">Index entries for sequences related to trinomials over GF(2)</a>

%t Select[Range[2, 1000], PrimitivePolynomialQ[x^# + x + 1, 2] &] (* _Robert Price_, Sep 19 2018 *)

%Y Cf. A002475, A073571, A057486.

%K nonn,nice,hard,more

%O 1,1

%A _Richard P. Brent_ and _Paul Zimmermann_, Sep 05 2002

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