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A001153 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.
(Formerly M0678 N0250)
6
2, 3, 5, 7, 17, 31, 89, 127, 521, 607, 1279, 2281, 3217, 4423, 9689, 19937, 23209, 44497, 110503, 132049, 756839, 859433, 3021377, 6972593, 24036583, 25964951, 30402457, 32582657, 42643801, 43112609 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also the list of "irreducible Mersenne trinomials" since here irreducible implies primitive.

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]

REFERENCES

Kurita, Yoshiharu and Matsumoto, Makoto; Primitive t-nomials (t=3,5) over GF(2) whose degree is a Mersenne exponent <= 44497. Math. Comp. 56 (1991), no. 194, 817-821.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. Zierler, On x^n+x+1 over GF(2). Information and Control 16 1970 502-505.

N. Zierler, Primitive trinomials whose degree is a Mersenne exponent. Information and Control 15 1969 67-69.

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.

LINKS

Table of n, a(n) for n=1..30.

Joerg Arndt, Matters Computational (The Fxtbook), see p.850 (but note errata for statement of Swan's theorem).

R. P. Brent, Searching for primitive trinomials (mod 2)

R. P. Brent, Trinomial Log Files and Certificates

R. P. Brent, Tables of trinomials (outdated)

R. P. Brent, S. Larvala and P. Zimmermann, A fast algorithm for testing reducibility of trinomials ..., Math. Comp. 72 (2003), 1443-1452.

Mathieu Ciet, Jean-Jacques Quisquater, Francesco Sica, A Short Note on Irreducible Trinomials in Binary Fields, in: 23rd Symposium on Information Theory in the BENELUX, Louvain-la-Neuve, Belgium, Macq, B., Quisquater, J.-J. (eds.), pp.233-234, (May-2002).

A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see p. 162.

Richard G. Swan, Factorization of polynomials over finite fields, Pacific Journal of Mathematics, vol.12, no.3, pp.1099-1106, (1962).

Index entries for sequences related to trinomials over GF(2)

CROSSREFS

Cf. A002475, A000043, A073571, A073639, A057486, A073726.

For smallest values of k, see A074743.

Sequence in context: A103383 A103382 A143027 * A141453 A100532 A231480

Adjacent sequences:  A001150 A001151 A001152 * A001154 A001155 A001156

KEYWORD

nonn,nice,hard,more

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Corrected and extended by Paul Zimmermann, Sep 05 2002.

Six more terms from Brent's page added by Max Alekseyev, Oct 22 2011

STATUS

approved

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Last modified November 28 15:24 EST 2014. Contains 250363 sequences.