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Support Bases of Solutions of a Functional Equation Arising From Multiplication of Quantum Integers and the Twin Primes Conjecture

  • Lan NguyenEmail author
Chapter

Summary

Let P be the support base of a solution Γ, with the field of coefficients of characteristic zero, of the functional equations arising from the multiplication of quantum integers discussed in [A. Borisov, M. Nathanson, Y. Wang, Quantum integers and cyclotomy, J. Number Theory (to appear); M. Nathanson, A functional equation arising from multiplication of quantum integers, J. Number Theory, 103(2), 214–233 (2003)]. It is known from the work of Nathanson as well as our work that there is a close relationship between P and the constructibility of Γ from quantum integers. In this paper, we prove that if the Twin Primes conjecture holds, then Γ is constructible from quantum integers if P contains infinitely many pairs of twin primes. This shows, in particular, that if the Twin Primes conjecture holds, then Γ is constructible from quantum integers whenever P has finite complement.

Keywords

Polynomial functional equations Q-series Quantum integers Quantum algebra 

References

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    Borisov, A., Nathanson, M., Wang, Y.: Quantum Integers and Cyclotomy, Journal of Number Theory, Volume 109, Issue 1, 120–135 (November 2004)MathSciNetGoogle Scholar
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    Nathanson, M.: A Functional Equation Arising From Multiplication of Quantum Integers, Journal of Number Theory, Volume 103, No. 2, 214–233 (2003)MathSciNetGoogle Scholar
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    Nguyen, L.: On the Classification of Solutions of a Functional Equation Arising from Multiplication of Quantum Integers (preprint)Google Scholar
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    Nguyen, L.: On the Solutions of a Functional Equation Arising from Multiplication of Quantum Integers. (To appear in Journal of Number Theory)Google Scholar
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    Nguyen, L.: On the Existence of Sequences of Polynomial Satisfying a Functional Equation Arising from Multiplication of Quantum Integers with a Given Support Base. (To appear in Journal of Number Theory)Google Scholar
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    Nguyen, L.: Extension of Supports of Solutions of a Functional Equation Arising from Multiplication of Quantum Integers (preprint)Google Scholar
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    Nguyen, L.: Solutions with Infinite Support Bases of a Functional Equation Arising from Multiplication of Quantum Integers (preprint)Google Scholar
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    Nguyen, L.: On the Polynomial and Maximal Solutions to a Functional Equation Arising from Multiplication of Quantum Integers (preprint)Google Scholar
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    Nguyen, L.: On the Rational Function Solutions of a Functional Equation Arising from Multiplication of Quantum Integers (preprint)Google Scholar
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    Nguyen, L.: On the Classification Rational Function Solutions of a Functional Equation Arising from Multiplication of Quantum Integers (preprint)Google Scholar
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    Nguyen, L.: On the Extension of Support of Rational Function Solutions of a Functional Equation Arising from Multiplication of Quantum Integers (preprint)Google Scholar
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    Nguyen, L.: Quantum Equivalence Relation of the set of Rational Function Solutions of a Functional Equation Arising from Multiplication of Quantum Integers (preprint)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Mathematics DepartmentThe University of MichiganAnn ArborUSA

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