A quantum number theory

We employ an algebraic procedure based on quantum mechanics to propose a `quantum number theory' (QNT) as a possible extension of the `classical number theory'. We built our QNT by defining pure quantum number operators (q-numbers) of a Hilbert space that generate classical numbers (c-numbers) belonging to discrete Euclidean spaces. To start with this formalism, we define a 2-component natural q-number textbf{N}, such that mathbf{N}² equiv N₁² + N₂², satisfying a Heisenberg-Dirac algebra, which allows to generate a set of natural c-numbers n in mathbb{N}. A probabilistic interpretation of QNT is then inferred from this representation. Furthermore, we define a 3-component integer q-number textbf{Z}, such that mathbf{Z}² equiv Z₁² + Z₂² + Z₃² and obeys a Lie algebra structure. The eigenvalues of each textbf{Z} component generate a set of classical integers m in mathbb{Z}cup frac{1}{2}mathbb{Z}^*, mathbb{Z}^* = mathbb{Z} setminus \{0\}, albeit all components do not generate mathbb{Z}³ simultaneously. We interpret the eigenvectors of the q-numbers as `q-number state vectors' (QNSV), which form multidimensional orthonormal basis sets useful to describe state-vector superpositions defined here as qunits. To interconnect QNSV of different dimensions, associated to the same c-number, we propose a quantum mapping operation to relate distinct Hilbert subspaces, and its structure can generate a subset W subseteq mathbb{Q}^*, the field of non-zero rationals. In the present description, QNT is related to quantum computing theory and allows dealing with nontrivial computations in high dimensions.