Lan Nguyen

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

Volume 18, 2012, Number 4, Pages 11—39

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## Details

### Authors and affiliations

Lan Nguyen

*Department of Mathematics, University of Wisconsin-Parkside
*

### Abstract

We resolve two questions posed by Melvyn Nathanson, YangWang, and Alex Borisov concerning solutions with coefficients in ℚ of the functional equations arising from multiplication of quantum integers. First, we determine the necessary and sufficient criteria for determining when a rational function solution to these functional equations contains only polynomials. Second, we determine the sets of primes *P* for which there exist maximal solutions Γ* _{P}* to these functional equations with support bases

*P*. We also give an explicit description of these maximal solutions.

### Keywords

- Diophantine equation
- Factorial
- Fibonacci
- Brocard-Ramanujan

### AMS Classification

- 11P99
- 11C08

### References

- Borisov, A., Nathanson, M. B., Wang, Y., Quantum Integers and Cyclotomy, Journal of Number Theory, Vol. 109, 2004, No. 1, 120–135.
- Nathanson, M. B. Formal Power Series Arising From Multiplication of Quantum Integers, DIMACs Series in Discrete Mathematics and Theoretical Computer Science, Vol. 64, 2004,145–157.
- Nathanson, M. B., A Functional Equation Arising From Multiplication of Quantum Integers,Journal of Number Theory, Vol. 103, 2003, No. 2, 214–233.
- Nguyen, L., On the Classification of Solutions of a Functional Equation Arising from Multiplication of Quantum Integers, Uniform Distribution Theory Journal (accepted).
- 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 Support Base of a Functional Equation Arising from Multiplication of Quantum Integers, Journal of Number Theory, Volume 130, 2010, No. 6, 1348–1373.

## Related papers

## Cite this paper

APANguyen, L. (2012). On the polynomial and maximal solutions to a functional equation arising from multiplication of quantum integers. Notes on Number Theory and Discrete Mathematics, 18(4), 11-39.

ChicagoNguyen, Lan. “On the Polynomial and Maximal Solutions to a Functional Equation Arising from Multiplication of Quantum Integers.” Notes on Number Theory and Discrete Mathematics 18, no. 4 (2012): 11-39.

MLANguyen, Lan. “On the Polynomial and Maximal Solutions to a Functional Equation Arising from Multiplication of Quantum Integers.” Notes on Number Theory and Discrete Mathematics 18.4 (2012): 11-39. Print.