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A132448
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First primitive polynomial over GF(2) of degree n, X^n suppressed.
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5
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1, 3, 3, 3, 5, 3, 3, 29, 17, 9, 5, 83, 27, 43, 3, 45, 9, 39, 39, 9, 5, 3, 33, 27, 9, 71, 39, 9, 5, 83, 9, 175, 83, 231, 5, 119, 63, 99, 17, 57, 9, 63, 89, 101, 27, 303, 33, 183, 113, 29, 75, 9, 71, 125, 71, 149, 45, 99, 123, 3, 39, 105, 3, 27, 27, 365, 39, 163
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OFFSET
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1,2
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COMMENTS
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More precisely: minimum value for X=2 of polynomials P[X] with coefficients in GF(2) such that X^n+P[X] is primitive. Applications include maximum-length linear feedback shift registers with efficient implementation in software.
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LINKS
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EXAMPLE
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a(11)=5, or 101 in binary, representing the GF(2)[X] polynomial X^2+1, because X^11+X^2+1 is primitive, contrary to X^11, X^11+1, X^11+X^1, X^11+X^1+1 and X^11+X^2.
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MATHEMATICA
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i2px[i_]:=If[i>1, BitAnd[i, 1]+i2px[BitShiftRight[i, 1]]x, i ]; s={1}; For[n=2, n<69, ++n, For[i=3, !PrimitivePolynomialQ[i2px[i]+x^n, 2], i+=2]; AppendTo[s, i]]; s (* Francois R. Grieu, Jan 15 2021 *)
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CROSSREFS
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2^n+a(n) is the smallest member of A091250 at least 2^n. A132447(n) = a(n)+2^n and gives the corresponding primitive polynomial. Cf. A132450, similar, with restriction to at most 5 terms. Cf. A132452, similar, with restriction to exactly 5 terms. Cf. A132454, similar, with restriction to minimal number of terms.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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