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Search: id:A059928
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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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+0
3
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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
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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It is expected that the sequence contains infinitely many primes. The heuristics for the Mersenne sequence can be adapted to show that approximately clogN of the first N terms should be prime. In the paper (Einsiedler, Everest, Ward) cited, we tested this against a lot of 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 (noe(AT)sspectra.com), Sep 15 2003
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REFERENCES
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M. Einsiedler, G. Everest, T. Ward, Primes in sequences associated to polynomials, LMS J. Comp. Math. 3 (2000), 15-29
G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London, 1999.
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LINKS
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Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
G. Everest and T. Ward, Primes in Divisibility Sequences
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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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CROSSREFS
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Cf. A060478, A087612.
Sequence in context: A110483 A010164 A006084 this_sequence A068452 A010163 A109012
Adjacent sequences: A059925 A059926 A059927 this_sequence A059929 A059930 A059931
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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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More terms from T. D. Noe (noe(AT)sspectra.com), Sep 15 2003
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