Additive Number Theory and the Ring of Quantum Integers

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.