Additive number theory and the ring of quantum integers

2002/03/31 by Melvyn B. Nathanson, Nathanson, Melvyn B.
Computer Science · Mathematics · #11B13 #30B12 #81R50 #Advanced Mathematical Identities #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT) #Numerical Methods and Algorithms #Quantum Algebra (math.QA) #math.NT #math.QA #msc:11B13 #msc:30B12 #msc:81R50

paper · pdf · doi:10.48550/arxiv.math/0204006

LaTex. 7 pages

arxiv created 2002/03/31 · openalex publication_date 2002/03/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ⊕_q [n]_q = [m+n]_q and a natural polynomial multiplication such that [m]_q ⊗_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.

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