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A073571
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Irreducible trinomials: numbers n such that x^n + x^k + 1 is an irreducible polynomial (mod 2) for some k with 0 < k < n.
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6
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2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 17, 18, 20, 21, 22, 23, 25, 28, 29, 30, 31, 33, 34, 35, 36, 39, 41, 42, 44, 46, 47, 49, 52, 54, 55, 57, 58, 60, 62, 63, 65, 66, 68, 71, 73, 74, 76, 79, 81, 84, 86, 87, 89, 90, 92, 93, 94, 95, 97, 98, 100, 102, 103, 105, 106, 108, 110, 111, 113
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OFFSET
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1,1
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COMMENTS
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This sequence is infinite: Golomb, "Shift Register Sequences," on p. 96 (1st ed., 1966) states that "It is easy to exhibit an infinite class of irreducible trinomials. viz. x^(2*3^a) + x^(3^a) + 1 for all a = 0, 1, 2, ..., but whose roots have only 3^(a+1) as their period." - A. M. Odlyzko, Dec 05 1997.
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REFERENCES
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S. W. Golomb, "Shift register sequence", revised edition, reprinted by Aegean Park Press, 1982. See Tables V-1, V-2.
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LINKS
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Table of n, a(n) for n=1..71.
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, Handbook of Applied Cryptography, CRC Press, 1996; see Table 4.6.
Index entries for sequences related to trinomials over GF(2)
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MAPLE
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a := proc(n) local k; for k from 1 to n-1 do if Irreduc(x^n+x^k+1) mod 2 then RETURN(n) fi od; NULL end: [seq(a(n), n=1..130)];
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MATHEMATICA
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irreducibleQ[n_] := (irr = False; k = 1; While[k < n, If[ Factor[ x^n + x^k + 1, Modulus -> 2] == x^n + x^k + 1, irr = True; Break[]]; k++]; irr); Select[ Range[120], irreducibleQ] (* Jean-François Alcover, Jan 07 2013 *)
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CROSSREFS
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For the numbers of such trinomials for a given n, see A057646.
See A073726 for primitive trinomials and A001153 for primitive Mersenne trinomials (and references). Complement of A057486. For values of k see A057774.
Sequence in context: A026470 A014133 A070115 * A191852 A136416 A072497
Adjacent sequences: A073568 A073569 A073570 * A073572 A073573 A073574
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KEYWORD
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nonn
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AUTHOR
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Paul Zimmermann, Sep 05 2002
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STATUS
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approved
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