Abstract
Let m and n be positive integers. For the quantum integer [n] q = 1+q+q 2+⋯+ q n − − 1 there is a natural polynomial addition such that [m] q ⊕ q [n] q = [m+n] q and a natural polynomial multiplication such that [m] q ⊗ q [n] q = [mn] q . These definitions are motivated by elementary decompositions of intervals of integers in combinatorics and additive number theory. This leads to the construction of the ring of quantum integers and the field of quantum rational numbers.
2000 Mathematics Subject Classification: Primary 11B75, 11N80, 05A30, 16W35, 81R50. Secondary 11B13. Key words and phrases. Quantum integers, quantum addition and multiplication, polynomial functional equations, additive bases.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Kassel, C.: Quantum Groups, Graduate Texts in Mathematics, vol. 155. Springer, New York (1995)
Nathanson, M.B.: A functional equation arising from multiplication of quantum integers. J. Number Theory 103(2), 214–233 (2003)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Nathanson, M.B. (2006). Additive Number Theory and the Ring of Quantum Integers. In: Ahlswede, R., et al. General Theory of Information Transfer and Combinatorics. Lecture Notes in Computer Science, vol 4123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11889342_28
Download citation
DOI: https://doi.org/10.1007/11889342_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-46244-6
Online ISBN: 978-3-540-46245-3
eBook Packages: Computer ScienceComputer Science (R0)