Additive number theory and the ring of quantum integers

DOI10.1016/J.ENDM.2005.07.061zbMATH Open1158.11304arXivmath/0204006OpenAlexW2172799746MaRDI QIDQ3439290

Melvyn B. Nathanson

Publication date: 29 May 2007

Published in: Electronic Notes in Discrete Mathematics (Search for Journal in Brave)

Abstract: Let $m$ and $n$ be positive integers. For the quantum integer $[n{]}_q=1+q+...+q^{n-1}$ there is a natural polynomial addition such that $[m{]}_{q}oplu{s}_q[n{]}_q=[m+n{]}_q$ and a natural polynomial multiplication such that $[m{]}_{q}otime{s}_q[n{]}_q=[mn{]}_q$. These constructions lead to the construction of the ring of quantum integers and the field of quantum rational numbers. It is also shown that addition and multiplication of quantum integers are equivalent to elementary decompositions of intervals of integers in additive number theory.

Full work available at URL: https://arxiv.org/abs/math/0204006

Mathematics Subject Classification ID

Binomial coefficients; factorials; (q)-identities (11B65) Elementary theory of partitions (11P81) Additive bases, including sumsets (11B13)

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