%I M0678 N0250 #64 Aug 06 2018 05:22:50
%S 2,3,5,7,17,31,89,127,521,607,1279,2281,3217,4423,9689,19937,23209,
%T 44497,110503,132049,756839,859433,3021377,6972593,24036583,25964951,
%U 30402457,32582657,42643801,43112609
%N Degrees of primitive irreducible trinomials: n such that 2^n - 1 is a Mersenne prime and x^n + x^k + 1 is a primitive irreducible polynomial over GF(2) for some k with 0 < k < n.
%C Also the list of "irreducible Mersenne trinomials" since here irreducible implies primitive.
%C Further terms of the form +-3 (mod 8) are unlikely, as the only possibility of an irreducible trinomial for n == +-3 (mod 8) is (by Swan's theorem) x^n+x^2+1 (and its reciprocal); see the Ciet et al. and the Swan reference. - _Joerg Arndt_, Jan 06 2014
%C The first Mersenne prime exponent not ruled out by Swan's theorem and yet not a member of this sequence is 57885161. - _Gord Palameta_, Jul 20 2018
%C 74207281 is also in the sequence. - _Gord Palameta_, Jul 20 2018
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, see p.850 (but note errata for statement of Swan's theorem).
%H Joerg Arndt, <a href="http://www.jjj.de/mathdata/all-trinomial-primpoly.txt">Complete list of primitive trinomials over GF(2) up to degree 400</a>.
%H Joerg Arndt, <a href="/A001153/a001153.txt">Complete list of primitive trinomials over GF(2) up to degree 400</a> [Cached copy, with permission]
%H R. P. Brent, <a href="http://maths.anu.edu.au/~brent/trinom-old.html">Searching for primitive trinomials (mod 2)</a> (first, "old" page)
%H R. P. Brent, <a href="http://maths.anu.edu.au/~brent/trinom.html">Searching for primitive trinomials (mod 2)</a> (second, current page)
%H R. P. Brent, <a href="http://maths.anu.edu.au/~brent/trinomlg.html">Trinomial Log Files and Certificates</a>
%H Richard P. Brent and Paul Zimmerman, <a href="https://hal.inria.fr/hal-01378493">Twelve New Primitive Binary Trinomials</a>, HAL Id : hal-01378493.
%H R. P. Brent, S. Larvala and P. Zimmermann, <a href="http://maths.anu.edu.au/~brent/pd/rpb199.pdf">A fast algorithm for testing reducibility of trinomials ...</a>, Math. Comp. 72 (2003), 1443-1452.
%H Mathieu Ciet, Jean-Jacques Quisquater, Francesco Sica, <a href="http://www.uclouvain.be/crypto/publications/year/2002">A Short Note on Irreducible Trinomials in Binary Fields</a>, in: 23rd Symposium on Information Theory in the BENELUX, Louvain-la-Neuve, Belgium, Macq, B., Quisquater, J.-J. (eds.), pp.233-234, (May-2002).
%H Yoshiharu Kurita and Makoto Matsumoto, <a href="http://dx.doi.org/10.1090/S0025-5718-99-01168-0">Primitive t-nomials (t=3,5) over GF(2) whose degree is a Mersenne exponent <= 44497</a>, Math. Comp. 56 (1991), no. 194, 817-821.
%H A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, <a href="http://www.cacr.math.uwaterloo.ca/hac/">Handbook of Applied Cryptography</a>, CRC Press, 1996; see p. 162.
%H Richard G. Swan, <a href="http://projecteuclid.org/euclid.pjm/1103036322">Factorization of polynomials over finite fields</a>, Pacific Journal of Mathematics, vol.12, no.3, pp.1099-1106, (1962).
%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>
%Y Cf. A002475, A000043, A073571, A073639, A057486, A073726.
%Y For smallest values of k, see A074743.
%K nonn,nice,hard,more
%O 1,1
%A _N. J. A. Sloane_
%E Corrected and extended by _Paul Zimmermann_, Sep 05 2002
%E Six more terms from Brent's page added by _Max Alekseyev_, Oct 22 2011