Lan Nguyen

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 20, 2014, Number 3, Pages 1—28

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## Details

### Authors and affiliations

Lan Nguyen

*Mathematics Department, University of Michigan-Ann Arbor
Ann Arbor, MI 48109, United States
*

### Abstract

It follows from our previous works and those of Nathanson that if *P* is a set of primes, then the greater the cardinality of *P*, the less likely that there exists a sequence of polynomials, satisfying the functional equation arising from multiplication of quantum integers studied by Nathanson, which has *P* as its support base and which cannot be generated by quantum integers. In this paper we analyze the set of roots of the polynomials involved leading to a direct construction of a polynomial solution Γ which has infinite support base *P* and which cannot be generated by quantum integers. Our results demonstrate that there are more to these solutions than those provided by quantum integers. In addition, we also show that a result of Nathanson does not hold if the condition *t*_{Γ} = 1 is removed.

### Keywords

- Diophantine equation
- Quantum integer
- q-series

Sumset - Polynomial functional equation
- Cyclotomic polynomial

### AMS Classification

- 11P99
- 11C08

### References

- Borisov, A., M. Nathanson, Y.Wang, Quantum Integers and Cyclotomy, Journal of Number Theory, Vol. 109, 2004, No. 1, 120–135.
- Nathanson, M., A Functional Equation Arising From Multiplication of Quantum Integers, Journal of Number Theory, Vol. 103, 2003, No. 2, 214–233.
- Nguyen, L., The Grothendieck Group associated to the Collection of all Solutions of a Functional Equation Arising from Multiplication of Quantum Integers with a given Support. Proceeding of Combinatorics and Additive Number Theory, 2011 (to appear).
- Nguyen, L., On the Solutions of a Functional Equation Arising from Multiplication of Quantum Integers, Journal of Number Theory Vol. 130, 2010, No. 6, 1292–1347.
- Nguyen, L., On the Classification of Solutions of a Functional Equation Arising from Multiplication of Quantum Integers. Uniform Distribution Theory Journal, 2012 (to appear).
- Nguyen, L., On the Support Base of a Functional Equation Arising from Multiplication of Quantum Integers, Journal of Number Theory, Vol. 130, 2010, Issue 6, 1348–1373.
- Nguyen, L., Extension of Supports of Solutions of a Functional Equation Arising from Multiplication of Quantum Integers. (in preparation)

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## Cite this paper

Nguyen, L. (2014). Solutions with infinite support bases of a functional equation arising from multiplication of quantum integers. Notes on Number Theory and Discrete Mathematics, 20(3), 1-28.