Author(s)

Tadashi Taniguchi

Faculty of Education, Gunma University, Maebashi, Japan.

Using elementary Mother space of type $e^M$ , we construct semiring $\mathcal{N}$ , ring $\mathcal{Z}$ and field $\mathcal{Q}$ of extended numbers inculuding natural numbers $\mathbb{N}$ , rational integers $\mathbb{Z}$ and rational numbers $\mathbb{Q}$ . We see that $\mathcal{N}$ and $\mathcal{Z}$ have a natural total order structure. They are also countable sets. It can be seen that the ring $\mathcal{Z}$ and the field $\mathcal{Q}$ are partially differentiable rings and fields. As an application, we construct a partially differentiable Riemann manifold over the Mother Pythagoras field $K$ . We also formulate the ABC type conjecture regarding $\mathcal{N}$ , $\mathcal{Z}$ , $\mathcal{Q}$ and their Mother algebraic extensions $L$ . And we introduce the Diophantine type equations related to the concept of this paper.

Monoid, Mother Number, Dark Number, Partially Differentiable Manifold, ABC Type Conjecture

Taniguchi, T. (2025) Arithmetic Differential Geometry over Mother Number Space. Advances in Pure Mathematics, 15, 775-814. doi: 10.4236/apm.2025.1512043.