Arithmetic Differential Geometry over Mother Number Space

Abstract

Using elementary Mother space of type e M , we construct semiring N , ring Z and field Q of extended numbers inculuding natural numbers ℕ , rational integers ℤ and rational numbers ℚ . We see that N and Z have a natural total order structure. They are also countable sets. It can be seen that the ring Z and the field 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 N , Z , Q and their Mother algebraic extensions L . And we introduce the Diophantine type equations related to the concept of this paper.