TITLE:
Arithmetic Differential Geometry over Mother Number Space
AUTHORS:
Tadashi Taniguchi
KEYWORDS:
Monoid, Mother Number, Dark Number, Partially Differentiable Manifold, ABC Type Conjecture
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.15 No.12,
December
29,
2025
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.