Additive number theory and the ring of quantum integers

Details Additive number theory and the ring of quantum integers Additive number theory and the ring of quantum integers

2002-03-31

arxiv.org/abs/math/0204006v1

Mathematics

Number Theory

LaTex. 7 pages

Scientific paper

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 $[m]_q \oplus_q [n]_q = [m+n]_q$ and a natural polynomial multiplication such that $[m]_q \otimes_q [n]_q = [mn]_q$. 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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