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A059928
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The periodic point counting sequence for the toral automorphism given by the polynomial of (conjectural) smallest Mahler measure. The map is x -> Ax mod 1 for x in [0,1)^10, where A is the companion matrix for the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.
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2
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1, 1, 1, 9, 1, 1, 1, 9, 1, 1, 1, 81, 1, 169, 841, 9, 1, 1, 1369, 9, 1, 529, 1, 81, 2401, 625, 1, 1521, 1, 841, 1024, 8649, 4489, 1, 5041, 729, 1, 1369, 6241, 9, 6889, 169, 29929, 4761, 841, 2209, 1, 178929, 85849, 2401, 10609, 5625, 100489, 2809, 11881, 1521, 1369
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OFFSET
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1,4
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COMMENTS
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It is expected that the sequence contains infinitely many squares of primes. The heuristics for the Mersenne sequence can be adapted to show that approximately c log N of the first N terms should be prime. The paper Einsiedler, Everest, Ward gives supporting numerical evidence.
The terms in this sequence are all squares. The sequence of square roots, A087612, is conjectured to contain an infinite number of primes. - T. D. Noe, Sep 15 2003
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REFERENCES
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G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.
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LINKS
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FORMULA
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The n-th term is abs(det(A^n-I)) where I is the 10 by 10 identity matrix and A is the companion matrix to the polynomial x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1.
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EXAMPLE
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The first term is 1 because Ax=x implies x=0 (since A-I) is invertible. Thus there is only one fixed point for the map.
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MATHEMATICA
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CompanionMatrix[p_, x_] := Module[{cl=CoefficientList[p, x], deg, m}, cl=Drop[cl/Last[cl], -1]; deg=Length[cl]; If[deg==1, {-cl}, m=RotateLeft[IdentityMatrix[deg]]; m[[ -1]]=-cl; Transpose[m]]]; c=CompanionMatrix[x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, x]; im=IdentityMatrix[10]; tmp=im; Table[tmp=tmp.c; Abs[Det[tmp-im]], {n, 100}] (From T. D. Noe)
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PROG
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(PARI) comp(pol) = my(v=Vec(pol), nn=poldegree(pol)); matrix(nn, nn, n, k, if (k==nn, -v[n], if(k==n-1, 1)));
id(nn) = matrix(nn, nn, n, k, n==k);
a(n) = my(p=x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1, m=comp(p)); abs(matdet(m^n-id(poldegree(p)))); \\ Michel Marcus, Nov 23 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Graham Everest (g.everest(AT)uea.ac.uk), Mar 01 2001
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EXTENSIONS
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STATUS
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approved
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