Just as the existence of unknown elementary particles is predicted in the world of elementary particles, there may also be unknown numbers in mathematics. The goal of this paper is to introduce and discuss unknown numbers. In this paper, this unknown number is called the dark numbers. Unknown numbers can be likened to dark matter in physics. The set of dark numbers and ordinary numbers is called the Mother numbers. The biggest feature is that a differential concept exists in the world of Mother numbers. That is, a Mother number can be differentiated by a dark number. Note that since ordinary numbers are included in the Mother number, a Mother number can not be differentiated by an ordinary number. Therefore, this differentiation is called a partial differentiation.

We will explain the reason for introducing the concept of Mother numbers. Kurokawa is studying absolute mathematics in order to challenge the Riemann hypothesis [1]. In particular, Kurokawa has studied the concept of absolute differentiation, which is differentiation by a prime element on a monoid [1]-[3]. Kurokawa’s absolute differentiation satisfies the Leibniz rule but does not satisfy linearity. We would like to make Kurokawa’s absolute differentiation also satisfy linearity. By extending the concept of number to Mother numbers, we can make it satisfy linearity. This also extends the Kurokawa’s category of monoid to the category of commutative ring.

The Mother number considered in this paper is a countable set, and therefore it is a discrete concept. Because of these properties, we want to develop continuous mathematics in discrete mathematics. This corresponds to turning classical physics into quantum physics. Let us explain the Mother number below.

Gotyou has described the concept of elementary Mother space [4]. Gotyou has explained that there are the following types of elementary Mother space:

In this paper, we will limit our discussion to the elementary Mother space of type e M .

Let us explain about elementary Mother space of type e M .

The generating function of the sequence { 1 n ! } n ∈ ℕ is

We substitute a set M for variable x . We also replace sum + with direct sum ∐ of sets, x n = x × ⋯ × x with direct product of sets M n = M × ⋯ × M , divide it in n ! with the action of permutation group S n in direct product of set M n . Hence we have

In this paper, this e M is called elementary Mother space.

We introduce the set D = { 1 , 2 , 3 , 4 , 5 , ⋯ } different from the usual set of natural numbers ℕ = { 1 , 2 , 3 , 4 , 5 , ⋯ } . However, 1 is a common, ordinary 1. We will call D the set of dark natural numbers. ( D , × ) is the monoid. Note that ( D , × ) and ( ℕ , × ) are monoid isomorphisms.

The elementary Mother space of D is the following.

For any unordered pair ( n 1 , n 2 , ⋯ , n l ) S l ∈ D l / S l , we define the element of D l / S l as follows.

By defining the commutative product × and the commutative sum + of the set e D , it becomes a commutative semiring ( e D , × , + ) (cf. Section 3).

We introduce the set D ^ = { ± 1 , ± 2 , ± 3 , ± 4 , ± 5 , ⋯ } different from the usual set of rational integers without zero ℤ \ { 0 } = { ± 1 , ± 2 , ± 3 , ± 4 , ± 5 , ⋯ } . We will call D ^ the set of dark rational integers. However, ±1 is a common, ordinary ±1. ( D ^ , × ) is the monoid. Note that ( D ^ , × ) and ( ℤ \ { 0 } , × ) are monoid isomorphisms.

The elementary Mother space of D ^ is the following.

For any unordered pair ( n 1 , n 2 , ⋯ , n l ) S l ∈ D ^ l / S l , we define the element of D ^ l / S l as follows.

By defining the commutative product × , sum + and difference − of the set e D ^ , it becomes a commutative ring ( e D ^ , × , + , − ) (cf. Section 3).

Each piece D l / S l and D ^ l / S l of e D and e D ^ is producing a new number n 1 ⊕ n 2 ⊕ ⋯ ⊕ n l .

Each piece of D l / S l and D ^ l / S l is only a set with new elements n 1 ⊕ n 2 ⊕ ⋯ ⊕ n l , but the set e D of the entire union of each piece is commutative semiring and the set e D ^ is commutative ring.

Furthermore, e D contains a natural total ordering structure ≤ . We write ( e D , ≤ , × , + , − ) , which is a totally order semiring. And e D ^ contains a natural total ordering structure ≤ . We write ( e D ^ , ≤ , × , + , − ) , which is a totally order ring (cf. Section 4 and 5). These are the new totally order semiring and totally order ring corresponding ( ℕ , ≤ , × , + ) and ( ℤ , ≤ , × , + , − ) .

The reason for considering e M is as follows. The coefficient 1 n ! that appears

in the expansion of e x corresponds to the fact that the permutation group S n is invariant with respect to e M . This means that the resulting ring is commutative. In this paper, we will only discuss commutative rings. Furthermore, although the mother number can be defined without using the notation of e M , using the notation of e M allows us to express various examples in a unified manner (cf. section 3).

Using e D as an example, we will explain what kind of number Mother number is,

According to this decomposition, e D is

Ordinary numbers in the set ℕ ∪ { 0 } are grouped as follows.

Dark numbers in the set e D \ ( ℕ ∪ { 0 } ) are grouped as follows.

Both dark prime and dark prime element are numbers that can not be factored into prime factors.

For example, it is

Primes in set e D of Mother numbers are grouped as follows.

For example, it is

One of the advantages of introducing e D and e D ^ is that they have the following properties.

There are monoid isomorphisms ( ℕ , × ) ≃ ( D , × ) , ( ℤ , × ) ≃ ( D ^ , × ) . Kurokawa’s absolute differential exists for these four monoids. Absolute differentiation satisfies the Leibniz rule but does not satisfy the linearity. It is a nonlinear differential (cf. [1]-[3]).

We want to linearly extend the nonlinear absolute differential on ( D , × ) . For this purpose, linear differentiation can be defined by extending the concept of number to e D . The same goes for ( D ^ , × ) (cf. Section 7).

We put ( N , ≤ , × , + ) : = ( e D , ≤ , × , + , − ) and ( Z , ≤ , × , + , − ) : = ( e D ^ , ≤ , × , + , − ) . Furthermore, the quotient field of the commutative ring Z is written as Q : = ( Z \ { 0 } ) − 1 Z .

In this paper, we will consider the following expansion of numbers.

We examine various arithmetic, algebraic and geometric properties of ( N , ≤ , × , + ) , ( Z , ≤ , × , + , − ) and ( Q , ≤ , × , ÷ , + , − ) . We also discuss the algebraic extension of Q and the algebraically closed field Q ¯ .

Each section of this paper is organized as follows.

Section 2 briefly reviews monoids and absolute algebras. Section 3 defines commutative semiring e D and commutative ring e D ^ . And we will introduce various examples of elementary Mother spaces and elementary Mother number spaces. In Sections 4 and 5, we rewrite e D and e D ^ as isomorphic objects to improve the clarity of the discussion. We also define a total order structure into them. Section 6 describes the fundamental theorem of number theory and the division theorem in e D ^ . In Section 7, we define the differential concept on e D , e D ^ and the Mother Pythagoras filed K . There are two important concepts of differentiation. One is point differentiation and the other is functional differentiation. Sections 8, 9 and 10 are the applications in this paper. In Section 8, we define partially differentiable Riemann manifolds with metric topology on K . In Section 9 and 10, we construct the ABC type conjectures about e D , e D ^ and the Mother algebraic extensions of Q denoted by L . We also introduce the Diophantine type equations related to the concept of this paper.

We explain the monoids and the absolute algebras by Kurokawa (cf. [1]-[3][5]).

Let ℕ = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , ⋯ } and ℤ = { 0 , ± 1 , ± 2 , ± 3 , ± 4 , ± 5 , ± 6 , ± 7 , ⋯ } be the set of all natural numbers and rational integers, respectively.

Definition 2.1.Amonoid A = ( A , × ) will be a semigroup with identity element 1 S .Anabsolute algebra A will be a monoid with zero element 0 A ,

Example 2.1. ( F 1 = { 1 } , × ) is the monoid.The corresponding absolute algebra is ( { 1 , 0 } , × ) .

Example 2.2. ( F 1 2 = { 1 , − 1 } , × ) is the monoid.The corresponding absolute algebra is ( { 1 , − 1 , 0 } , × ) .

Example 2.3. ( F 1 3 = { 1 , 1 2 ( − 1 + − 3 ) , 1 2 ( − 1 − − 3 ) } , × ) is the monoid.The corresponding absolute algebra is ( { 1 , 1 2 ( − 1 + − 3 ) , 1 2 ( − 1 − − 3 ) , 0 } , × ) .

Example 2.4. ( F 1 4 = { 1 , − 1 , i , − i } , × ) is the monoid.The corresponding absolute algebra is ( { 1 , − 1 , i , − i , 0 } , × ) .

In general, F 1 n is considered (cf. [1]).

Example 2.5. ( ℕ , × ) is the monoid and ( ℤ , × ) is the absolute algebra.

Kurokawa has defined the following interesting monoid.

Example 2.6.The 2 , 3 , 5 , 7 , ⋯ are ordinary primes.Then we define

If i 1 = i 2 = i 3 = ⋯ = 0 then 1 ∈ F 1 [ 2 , 3 , 5 , 7 , ⋯ ] .Hence ( F 1 [ 2 , 3 , 5 , 7 , ⋯ ] , × ) becomes the monoid.

Example 2.7.The 2 , 3 , 5 , 7 , ⋯ are ordinary primes.Then we define

If i 0 = i 1 = i 2 = i 3 = ⋯ = 0 then 1 ∈ F 1 [ − 1 , 0 , 2 , 3 , 5 , 7 , ⋯ ] .Hence ( F 1 [ − 1 , 0 , 2 , 3 , 5 , 7 , ⋯ ] , × ) becomes the absolute algebra.

Then the following important monoid isomorphic expression holds.

Let V be a vector space over a field k . Let S * ( V ) be a symmetric tensor algebra. e M , defined below, is similar to S * ( V ) . The difference is that M can be a general set. Also, products and sums are different. The product of S * ( V ) is a tensor product, and the sum is a direct sum.

Elementary Mother space is defined as follows [4].

where S n is the permutation group and we can set ∅ = 0 and S n M = ( M × M × ⋯ × M ) ︸ n   times / S n .

For example, ( M × M × M ) / S 3 = S 3 M means for ( x , y , z ) ∈ M × M × M , it is as follows.

The following example is from Nakajima [6].

Definition 3.1.We define the set D = { 1 , 2 , 3 , 4 , 5 , ⋯ } as the set of dark natural numbers and also define the set D ^ = { ± 1 , ± 2 , ± 3 , ± 4 , ± 5 , ⋯ } as the set of dark rational integers.

±1 are common to ordinary natural numbers and rational integers.

For the product × of elements of D , ( D , × ) is the monoid.

For the product × of elements of D ^ , ( D ^ , × ) is the monoid.

Below, we define two examples of the most basic elementary Mother number space in this paper.

The elementary Mother number space of D is defined as follows.

For any unordered pair ( n 1 , n 2 , ⋯ , n l ) S l ∈ D l / S l , we define the element of D l / S l as follows.

By defining the product and sum of the set e D as follows, it becomes a commutative semiring ( e D , × , + ) .

for any ( n 1 ⊕ ⋯ ⊕ n l ) ∈ D l / S l , ( m 1 ⊕ ⋯ ⊕ m k ) ∈ D k / S k .

The identity element for the product × is 1 ∈ D ^ .

The identity element for sum + is 0 ∈ { ∅ } .

The elementary Mother number space of D ^ is defined as follows.

For any unordered pair ( n 1 , n 2 , ⋯ , n l ) S l ∈ D ^ l / S l , we define the element of D ^ l / S l as follows.

By defining the product, sum and difference of the set e D ^ as follows, it becomes a commutative ring ( e D ^ , × , + , − ) .

for any ( n 1 ⊕ ⋯ ⊕ n l ) ∈ D ^ l / S l , ( m 1 ⊕ ⋯ ⊕ m k ) ∈ D ^ k / S k .

The identity element for the product × is 1 ∈ D .

The identity element for sum + is 0 ∈ { ∅ } .

The inverse of n 1 ⊕ ⋯ ⊕ n l with respect to the sum + is ⊖ n 1 ⊖ ⋯ ⊖ n l .

Each piece D ^ l / S l and D l / S l of e D and e D ^ is only a set with a new number n 1 ⊕ n 2 ⊕ ⋯ ⊕ n l , but the set e D of the entire union of each piece is a commutative semiring, and the set e D ^ is a commutative ring. We also define as follows.

and

This generally turns out to be

Proposition 3.1. ℕ ⊂ N is asubsemiringand ℤ ⊂ Z is a subring.

Proof. Let us consider that ℕ is a subset of N . If it can be shown that it is a subset, it is trivial that it is subsemiring. For any l , the special element D l / S l ∋ ( 1 , 1 , ⋯ , 1 ) S l = 1 ⊕ 1 ⊕ ⋯ ⊕ 1 can be calculated and 1 ⊕ 1 ⊕ ⋯ ⊕ 1 = 1 + 1 + ⋯ + 1 = l , and l is in ℕ . Therefore, ℕ is a subset of N . The same applies to ℤ ⊂ Z . □

Below, we will discuss various examples of elementary Mother spaces other than those mentioned above.

For the monoid ( A = { ± 1 , ± p , ± p 2 , ± p 3 , ⋯ } , × ) , it can be expressed as follows.

This is written as ( ℤ [ p ] , × , + , − ) .

For the monoid ( B = { ± p i 1 × q i 2 | i 1 , i 2 ∈ ℕ ∪ { 0 } , p ≠ q } , × ) , it can be expressed as follows.

This is written as ( ℤ [ p , q ] , × , + , − ) .

As we will see in section 5, ℤ [ p ] and ℤ [ p , q ] are subrings of e D ^ .

The following eight examples are obtained by taking the mother number e M of a known monoid M for a known commutative ring or commutative semiring.

For the monoid ( F 1 = { 1 } , × ) , it can be expressed as follows.

The following semiring isomorphism holds

For the monoid ( F 1 2 = { 1 , − 1 } , × ) , it can be expressed as follows.

The following ring isomrphism holds

For the monoid ( F 1 3 = { 1 , 1 2 ( − 1 + − 3 ) , 1 2 ( − 1 − − 3 ) } , × ) , it can be expressed as follows.

The following ring isomorphism holds

For the monoid ( F 1 4 = { 1 , − 1 , i , − i } , × ) , it can be expressed as follows.

The following ring isomorphism holds

Examples of the e F 1 and e F 1 2 show that F 1 and F 1 2 are hidden in the underground structure of ℕ ∪ { 0 } and ℤ . This is an example of the importance of Kurokawa’s philosophy of absolute mathematics (cf. [1]).

For the monoid ( ℤ * , × ) : = ( ℤ \ { 0 } , × ) , it can be expressed as follows.

The following ring isomorphism holds

For the monoid ( ℕ , × ) , it can be expressed as follows.

The following semiring isomorphism holds

Let us introduce yet another example.

A polynomial ring with one variable or infinite variables can also be constructed as an elementary Mother space as shown below.

For the monoid ( X = { ± 1 , ± X , ± X 2 , ± X 3 , ⋯ } , × ) of indefinite element X , it can be expressed as follows.

The following ring isomorphism holds

For the monoid ( Y = { ± X 1 i 1 X 2 i 2 X 3 i 3 ⋯ | i 1 , i 2 , i 3 , ⋯ ∈ ℕ ∪ { 0 } } , × ) with infinite number of indefinite elements X 1 , X 2 , X 3 , ⋯ , the following ring isomorphism holds.

Furthermore, the elementary Mother spaces of groups ℚ \ { 0 } , ℝ \ { 0 } and ℂ \ { 0 } with respect to the product are respectively ℚ , ℝ and ℂ which are fields.

As we can see from these examples, it can be seen that the monoid before taking the elementary Mother space representation is an extremely basic object. By taking the elementary Mother space representation of the monoid, natural sums and differences can be added. In order to obtain a new number concept, we introduced new monoids D = { 1 , 2 , 3 , 4 , 5 , ⋯ } and D ^ = { ± 1 , ± 2 , ± 3 , ± 4 , ± 5 , ⋯ } .

Let us introduce another interesting example of elementary Mother space.

For monoid A , I is a monoid ideal if a ∈ A , x ∈ I then a x ∈ I .

Monoid ideal I ⊊ A is a monoid prime ideal if a , b ∉ I then a b ∉ I .

Kurokawa thought of the following [5].

Spec ( ℤ , × ) is a set of monoid prime ideals of monoid ( ℤ , × ) . Then the monoid prime ideals of ( ℤ , × ) are

and we know that dim Spec ( ℤ , × ) = ∞ .

Lemma 3.1.For the set P = { 2 , 3 , 5 , 7 , 11 , ⋯ } of ordinary primes,elementary Mother space is

Then we have

Proof. We identify the following.

ℙ ∋ p ≃ ( p ) , ℙ 2 / S 2 ∋ ( p , q ) ≃ ( p ) ∪ ( q ) , in particular ( p , p ) ≃ ( p ) ∪ ( p ) ≃ ( p ) ∈ ℙ , ℙ 3 / S 3 ∋ ( p , q , r ) ≃ ( p ) ∪ ( q ) ∪ ( r ) , in particular ( p , p , r ) ≃ ( p ) ∪ ( p ) ∪ ( r ) ≃ ( p ) ∪ ( r ) ∈ ℙ 2 / S 2 and ( p , p , p ) ≃ ( p ) ∪ ( p ) ∪ ( p ) ≃ ( p ) ∈ P . The same applies below.

We define the sum + as follows.

The identity element for the sum is ( 0 ) . Therefore, we have ( e ℙ , + ) ≃ ( Spec ( ℤ , × ) , + ) . This is a Zariski topological monoid for the sum. □

On the other hand, Spec ( ℤ , × , + , − ) is only a Zariski topological space and does not contain a monoid structure for the sum.

In this section we will give an expression equivalent to ( e D , × , + ) . It turns out that ( e D , × , + ) is a polynomial semiring. However, it can be seen that total order structure is different.

Definition 4.1

where a i 1 i 2 i 3 ⋯ except for a finite number is zero.

From now on, ∑ finite sum will be abbreviated as ∑ .

For the set ℕ [ 2 , 3 , 5 , ⋯ ] , the product × and the sum + are defined as follows.

Hence ( ℕ [ 2 , 3 , 5 , ⋯ ] , × , + ) becomes the commutative semiring.

Note that a i 1 i 2 i 3 ⋯ ⊕ b i 1 i 2 i 3 ⋯ can be calculated and a i 1 i 2 i 3 ⋯ ⊕ b i 1 i 2 i 3 ⋯ = a i 1 i 2 i 3 ⋯ + b i 1 i 2 i 3 ⋯ = n , and n is an ordinary number. On the other hand, for example, 3 ⊕ 5 is a single number and can not be calculated any further.

Lemma 4.1.The following semiring isomorphism holds.

Proof.

where, p 1 , p 2 , ⋯ , p n , q 1 , q 2 , ⋯ , q m , r 1 , r 2 , ⋯ , r u are dark primes and j 1 ≥ 0 , j 2 ≥ 0 , ⋯ , j n ≥ 0 , k 1 ≥ 0 , k 2 ≥ 0 , ⋯ , k m ≥ 0 , l 1 ≥ 0 , l 2 ≥ 0 , ⋯ , l u ≥ 0 , i 1 ≥ 0 , i 2 ≥ 0 , i 3 ≥ 0 , ⋯ . ⋯ is finite. □