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Search: id:A071642
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| A071642 |
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Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(2). |
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8
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| 0, 1, 2, 4, 10, 12, 18, 28, 36, 52, 58, 60, 66, 82, 100, 106, 130, 138, 148, 162, 172, 178, 180, 196, 210, 226, 268, 292, 316, 346, 348, 372, 378, 388, 418, 420, 442, 460, 466, 490, 508, 522, 540, 546, 556, 562, 586, 612, 618, 652, 658, 660, 676, 700, 708, 756, 772
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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All polynomials of odd degree > 1 are reducible over GF(2).
For k > 2, a(k) = A001122(k-2) - 1 due to the relationship between cycles and irreducibility. - T. D. Noe (noe(AT)sspectra.com), Sep 09 2003
For the values of n in the sequence there exists a type-1 optimal normal basis over GF(2). The corresponding field polynomial is the all-ones polynomial x^n+x^(n-1)+...+1. - Joerg Arndt (arndt(AT)jjj.de), Feb 25 2008
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REFERENCES
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M. Olofsson, VLSI Aspects on Inversion in Finite Fields, Dissertation No. 731, Dept. Elect. Engin., Linkoping, Sweden, 2002.
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LINKS
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Joerg Arndt, Fxtbook
Eric Weisstein's World of Mathematics, Irreducible Polynomial
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MATHEMATICA
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Do[s = Sum[x^i, {i, 0, n}]; If[ ToString[ Factor[s, Modulus -> 2]] == ToString[s], Print[n]], {n, 2, 1000, 2}]
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CROSSREFS
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Cf. A001122 (primes with primitive root 2).
Sequence in context: A085344 A047463 A107059 this_sequence A034166 A092367 A127591
Adjacent sequences: A071639 A071640 A071641 this_sequence A071643 A071644 A071645
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KEYWORD
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easy,nonn
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AUTHOR
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njas Jun 22 2002
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EXTENSIONS
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Extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 24 2002
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