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Edit: refined the description of $S$.

As many readers said that the condition (3) is vague, I replace it by a more concrete but alternative condition (3'). Keep in mind that this restricts the possibilities:

(3') There is an addition on $S$ making $S$ a semiring and $f$ an embedding of semirings. Moreover,

• (3'a) the set $E$ is countably infinite
• (3'b) there is a polynomial $g \in S[x]$ such that the map $g \circ f: \mathbb N^+ \to E$ is a bijection. (This says that there is a polynomial formula for the new primes $E$)

Edit: refined the description of $S$. Added an alternative condition to (3).

Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The indecomposable ones, that cannot be further factored, are called primes. But the primes are irregular and the most difficult to understand. So, instead of trying hard to find some pattern of primes, why not consider finding an extension of the integer system so that primes can be further factored into another ones that admit some regularity?

Question: Let $\mathbb N^+$ be the set of positive integers equipped with its canonical addition $+$, multiplication $\times$, and total order $\leq$. Let $\mathbb P$ be the set of all primes. Is there any related work concerning finding (or proving non-existence of) an algebraic system $S$ such that:

(0) there is an binary operation $\cdot : S\times S \to S$ such that $(S,\cdot)$ is a commutative monoid.

(1) there is a monoid embedding $f: (\mathbb N^+, \times) \to (S, \cdot)$.

(2) there is a subset $E := \{e_1, \ldots,e_n, \ldots \} \subset S$ such that

• (2a) any $e \in E$ is not a unit.

• (2b) if $e = a \cdot b$ for any $e \in E$ and $a,b \in S$, then either $a$ or $b$ is a unit.

• (2c) for any $p \in \mathbb P$, the element $f(p)$ has factorization $f(p) = \prod_{e \in E} e^{x_e}$ for $x_e \in \mathbb N$ and this factorization is unique up to a unit.

(3) the structure of $E$ in $S$ is simpler than the structure of $\mathbb P$ in $\mathbb N^+$. (In other words, primes are better understood in the language of $E$. This requires $S$ to have more structure than a monoid)

Edit: refined the description of $S$.

Note that the extension of the addition operation is not considered yet, so the system we are searching for need not be a ring. It is easy to find some system satisfying (1) and (2). For example, the ring $\mathbb Z[{\sqrt{2}}]$, but factoring numbers to $\{\sqrt{2}, 3, 5 \ldots\}$ doesn't make the situation easier, so the condition (3) is not satisfied. The ring of rationales $\mathbb Q$ doesn't simultaneously satisfy (2) and (3).

I don't expect that any such system can be found easily; since if such one is found, then the study of primes immediately becomes obsolete. Instead, I'd like to discover related researching field and potentially useful connections from established results. So any help will be appreciated. Thanks in advance.

As an aside, the set of indecomposable elements $E$ cannot be finite. Here is the reason. Assume that there exists a such system with the cardinality of $E$ being $n < \infty$, then by factoring each prime number into products of $e_i \in E$, we associates each prime number to a vector in the $n$-dimensional $\mathbb Z$-module $\mathbb Z^n$, by sending a prime $p$ to $(x_1,\ldots,x_n)$ where $x_i$ is the multiplicity of $e_i$ in the factorization of $p$. Then any $n+1$ many primes are associated to $n+1$ vectors, which are linear dependent. By identifying the addition in that vector space with the multiplication of associated numbers, we obtain that for any $n+1$ primes, some power of some of them are powers of other primes, a contradiction.

This also gives me inspiration. By considering all primes as a set of linear independent vectors in a infinite dimensional $\mathbb Z$-space, the whole natural numbers is considered as its span. Adding the canonical basis into this span is exactly one way to extend the number system.

Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The indecomposable ones, that cannot be further factored, are called primes. But the primes are irregular and the most difficult to understand. So, instead of trying hard to find some pattern of primes, why not consider finding an extension of the integer system so that primes can be further factored into another ones that admit some regularity?

Question: Let $\mathbb N^+$ be the set of positive integers equipped with its canonical addition $+$, multiplication $\times$, and total order $\leq$. Let $\mathbb P$ be the set of all primes. Is there any related work concerning finding (or proving non-existence of) an algebraic system $S$ such that:

(0) there is an binary operation $\cdot : S\times S \to S$ such that $(S,\cdot)$ is a commutative monoid.

(1) there is a monoid embedding $f: (\mathbb N^+, \times) \to (S, \cdot)$.

(2) there is a subset $E := \{e_1, \ldots,e_n, \ldots \} \subset S$ such that

• (2a) any $e \in E$ is not a unit.

• (2b) if $e = a \cdot b$ for any $e \in E$ and $a,b \in S$, then either $a$ or $b$ is a unit.

• (2c) for any $p \in \mathbb P$, the element $f(p)$ has a unique non-trivial factorization $f(p) = \prod_{e \in E} e^{x_e}$ for $x_e \in \mathbb N$.

(3) the structure of $E$ in $S$ is simpler than the structure of $\mathbb P$ in $\mathbb N^+$. (In other words, primes are better understood in the language of $E$. This requires $S$ to have more structure than a monoid)

Edit: refined the description of $S$.

Note that the extension of the addition operation is not considered yet, so the system we are searching for need not be a ring. It is easy to find some system satisfying (1) and (2). For example, the ring $\mathbb Z[{\sqrt{2}}]$, but factoring numbers to $\{\sqrt{2}, 3, 5 \ldots\}$ doesn't make the situation easier, so the condition (3) is not satisfied. The ring of rationales $\mathbb Q$ doesn't simultaneously satisfy (2) and (3).

I don't expect that any such system can be found easily; since if such one is found, then the study of primes immediately becomes obsolete. Instead, I'd like to discover related researching field and potentially useful connections from established results. So any help will be appreciated. Thanks in advance.

As an aside, the set of indecomposable elements $E$ cannot be finite. Here is the reason. Assume that there exists a such system with the cardinality of $E$ being $n < \infty$, then by factoring each prime number into products of $e_i \in E$, we associates each prime number to a vector in the $n$-dimensional $\mathbb Z$-module $\mathbb Z^n$, by sending a prime $p$ to $(x_1,\ldots,x_n)$ where $x_i$ is the multiplicity of $e_i$ in the factorization of $p$. Then any $n+1$ many primes are associated to $n+1$ vectors, which are linear dependent. By identifying the addition in that vector space with the multiplication of associated numbers, we obtain that for any $n+1$ primes, some power of some of them are powers of other primes, a contradiction.

This also gives me inspiration. By considering all primes as a set of linear independent vectors in a infinite dimensional $\mathbb Z$-space, the whole natural numbers is considered as its span. Adding the canonical basis into this span is exactly one way to extend the number system.

Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The indecomposable ones, that cannot be further factored, are called primes. But the primes are irregular and the most difficult to understand. So, instead of trying hard to find some pattern of primes, why not consider finding an extension of the integer system so that primes can be further factored into another ones that admit some regularity?

Question: Is there any related work concerning finding (or proving non-existence of) an algebraic system $S$ such that:

(1) $S$ contains the positive integers $\mathbb N^+$

(2) the multiplication of integers is extended to this system and is still associative and commutative. Moreover, any positive integer (equivalently, prime number) is a unique product of a set of indecomposable elements $E := \{e_1, \ldots,e_n, \ldots \} \subset S$.

(3) the structure of $E$ is simpler than the structure of all primes.

Note that the extension of the addition operation is not considered yet, so the system we are searching for is not necessarily to be a ring. It is easy to find some system satisfying (1) and (2). For example, the ring $\mathbb Z[{\sqrt{2}}]$, but factoring numbers to $\{\sqrt{2}, 3, 5 \ldots\}$ doesn't make the situation easier, so the condition (3) is not satisfied. The ring of rationals $\mathbb Q$ doesn't simultaneously satisfy (2) and (3).

I don't expect that any such system can be found easily; since if such one is found, then the study of primes immediately becomes obsolete. Instead, I'd like to discover related researching field and potentially useful connections from established results. So any help will be appreciated. Thanks in advance.

As an aside, the set of indecomposable elements $E$ cannot be finite. Here is the reason. Assume that there exists a such system with the cardinality of $E$ being $n < \infty$, then by factoring each prime number into products of $e_i \in E$, we associates each prime number to a vector in the $n$-dimensional $\mathbb Z$-module $\mathbb Z^n$, by sending a prime $p$ to $(x_1,\ldots,x_n)$ where $x_i$ is the multiplicity of $e_i$ in the factorization of $p$. Then any $n+1$ many primes are associated to $n+1$ vectors, which are linear dependent. By identifying the addition in that vector space with the multiplication of associated numbers, we obtain that for any $n+1$ primes, some power of some of them are powers of other primes, a contradiction.

This also gives me inspiration. By considering all primes as a set of linear independent vectors in a infinite dimensional $\mathbb Z$-space, the whole natural numbers is considered as its span. Adding the canonical basis into this span is exactly one way to extend the number system.

Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The indecomposable ones, that cannot be further factored, are called primes. But the primes are irregular and the most difficult to understand. So, instead of trying hard to find some pattern of primes, why not consider finding an extension of the integer system so that primes can be further factored into another ones that admit some regularity?

Question: Let $\mathbb N^+$ be the set of positive integers equipped with its canonical addition $+$, multiplication $\times$, and total order $\leq$. Let $\mathbb P$ be the set of all primes. Is there any related work concerning finding (or proving non-existence of) an algebraic system $S$ such that:

(0) there is an binary operation $\cdot : S\times S \to S$ such that $(S,\cdot)$ is a commutative monoid.

(1) there is a monoid embedding $f: (\mathbb N^+, \times) \to (S, \cdot)$.

(2) there is a subset $E := \{e_1, \ldots,e_n, \ldots \} \subset S$ such that

• (2a) any $e \in E$ is not a unit.

• (2b) if $e = a \cdot b$ for any $e \in E$ and $a,b \in S$, then either $a$ or $b$ is a unit.

• (2c) for any $p \in \mathbb P$, the element $f(p)$ has factorization $f(p) = \prod_{e \in E} e^{x_e}$ for $x_e \in \mathbb N$ and this factorization is unique up to a unit.

(3) the structure of $E$ in $S$ is simpler than the structure of $\mathbb P$ in $\mathbb N^+$. (In other words, primes are better understood in the language of $E$. This requires $S$ to have more structure than a monoid)

Edit: refined the description of $S$.

Note that the extension of the addition operation is not considered yet, so the system we are searching for need not be a ring. It is easy to find some system satisfying (1) and (2). For example, the ring $\mathbb Z[{\sqrt{2}}]$, but factoring numbers to $\{\sqrt{2}, 3, 5 \ldots\}$ doesn't make the situation easier, so the condition (3) is not satisfied. The ring of rationales $\mathbb Q$ doesn't simultaneously satisfy (2) and (3).

I don't expect that any such system can be found easily; since if such one is found, then the study of primes immediately becomes obsolete. Instead, I'd like to discover related researching field and potentially useful connections from established results. So any help will be appreciated. Thanks in advance.

As an aside, the set of indecomposable elements $E$ cannot be finite. Here is the reason. Assume that there exists a such system with the cardinality of $E$ being $n < \infty$, then by factoring each prime number into products of $e_i \in E$, we associates each prime number to a vector in the $n$-dimensional $\mathbb Z$-module $\mathbb Z^n$, by sending a prime $p$ to $(x_1,\ldots,x_n)$ where $x_i$ is the multiplicity of $e_i$ in the factorization of $p$. Then any $n+1$ many primes are associated to $n+1$ vectors, which are linear dependent. By identifying the addition in that vector space with the multiplication of associated numbers, we obtain that for any $n+1$ primes, some power of some of them are powers of other primes, a contradiction.

This also gives me inspiration. By considering all primes as a set of linear independent vectors in a infinite dimensional $\mathbb Z$-space, the whole natural numbers is considered as its span. Adding the canonical basis into this span is exactly one way to extend the number system.